mathhmagic.blogspot.gr : ., ., ., ., . 0

mathhmagic.blogspot.gr : ., ., ., ., . 1 . , , , . , ., viral, on line . , . . . . !! .... 380 . g(x)F'(x)= f(t)dt ' !! ? ? ? ? ? ? ??????? ? ? ? ? ? ? ? ? ? ????

mathhmagic.blogspot.gr : ., ., ., ., . 2 1) , ,,.... 2) , ., 3) , ., 4) , , 5) , ., 6) , ., 7) , ., 8) , . , 9) , , 10) ,. . , 11)-, . , 12)-, .. , 13)-, .., 14) 1,2,3, ., 15) 1000+1 , ., 16) , & ., 17) , ...., , 18) 19), . 20) . ,., 21) ,-, 22) .. . 23),., 24), . 25) , Spivak M. ,. 26) . , 27)Problems in Calculus ,..Maron,Mir Publisher 28) , ., 29) 30) , , 31) , ., 32) , . ., . , . , . 33)Problem book:Algebra and Elementary functions, Kutepov A.,Rubanov, MIR Publishers 34) , 35)The theory of functions of a real variable, R.L.Jeffery 36)A Problem book in mathematical analysis,G.N Berman 37) Bad problems in Calculus, A.G .Drolkun 38) 1,2,3 .,. 39) , ., 40) , .. 41) ,. 42) . 43) , 44)Differential Calculus ,.Ball 45)Calculus,E.Swokowski 46)Problems in Algebra ,T.Andreesku, Z.Feng 47) , .. . 48)O (Lisari team) 49) , , ( qr-code) .

mathhmagic.blogspot.gr : ., ., ., ., . 3 : 50 4 : 151 37 : 50 164 : 110 233 (,,,) ...: 20 329 434 f f(x)=

mathhmagic.blogspot.gr : ., ., ., ., . 4- 1) f : : f(x y) f(x) f(y) xy+ = + x, y x 0f(x)lim 10x= . x 2f(x) f(2)limx 2. x 2 y = x y 2= + . x 2 y 0 . : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )x 2 y 0 y 0 y 0(1)y 0 y 0 y 0f y 2 f 2 f y f 2 y 2 f 2 f y 2yf x f 2lim lim lim limx 2 y y yf y f y f y2ylim lim 2 lim 2 10 2 8.y y y y + + = = = = = = = = 2) + =332x 4x 1974 xf(x)x 2015, . xlim f(x) . + = = 3 3x xlim(2x 4x 1974) lim(2x ) . . ( ) x , + + > <

mathhmagic.blogspot.gr : ., ., ., ., . 5( )(x 3) 6 26x 18 2x 6 6(x 3) 2(x 3) 6 2....(x 1)(x 3) (x 1)(x 3) (x 1)(x 3) x 1 + + = = = = x 3 x 36 2 6 2lim f(x) lim 3 1x 1 2 = = = : 3 1 8 3 = = (1) = + = + = + = = 35 2 5( 3) 2 15 2 13 13= , 3= x 3lim f(x) 8= . ii) ) 13= , 3= 2 2 22x 0 x 0 x 0 x 013 14 3 4 1 1 1 1 1 1lim lim lim limx x x xx x x x + + = = = = x 01 1lim (1 ) ( )(1 ( )) ( )( )x x = = + + = + = ) 3, 13x x x 5 x 3 5 x 2 xlim lim limx 14 x 13 14 x 1= = + + = =+ + + + : ( )x lim x 1 1 1 0+ = + = 3x , ,2 2 : x 1 x 1 0> + > x 1limx 1= ++ x x x x 2 x 1 1lim lim 2 x lim 2 x lim 2 ( )x 1 x 1 x 1 = = = + = ++ + + ) = = = = + + + + x 0 x 0 x 0 x 0x x x xlim lim lim limx 4 1 x ( 13) 4( 3) 1 x 13 12 1 x 1 1 ( )( )( )( )( )( )( )2x 0 x 0 x 0 x 0 2x x 1 1 x x 1 1x x x x xxlim lim lim limx x xx 1 1x 1 1 x x 1 1 x 1 1 x 1 1 + + + + = = = + + + + + + ( ) ( ) ( ) ( )x 0 x 0 x 0 x 0 x 0x x 1 1 x x 1 1x x x xlim lim lim x 1 1 lim lim x 1 1x x 1 1 x x x x + + + + = = + + = + + + 1 2 2= = ..!!

mathhmagic.blogspot.gr : ., ., ., ., . 64) f )0, + : 3f(x) xf(x) 1 + = f(x) 0> x 0 . ( )x 0lim f x 1+= . : ( )x 0lim f(x) 1 0+ = . . : ( )[ ] ( ) ( )[ ] ( )( ) ( ) ( )( )[ ] ( )( ) ( ) ( )[ ] ( )[ ] ( )( ) ( )3 23 22 2f x 1 xf x 0 f x 1 f x f x 1 1 xf x x x 0f x 1 f x f x 1 x f x 1 x f x 1 f x f x 1 x x + = + + + + = + + + = + + + = ( )x 0,f x 0 > ( )( ) ( )2f x f x x 1 0+ + + > , ( )( )( ) ( )2xf x 1f x f x x 1 =+ + +, ( )( )( ) ( )2xf x 1 xf x f x x 1 = + + + : ( ) ( )x f x 1 x 1 x f x 1 x + ( ) ( )x 0 x 0lim 1 x lim 1 x 1+ + = + = , : ( )x 0lim f x 1+= 5) f : + = +f(x)f(y) xy xf(y) yf(x) , x, y i) f. ii) 3x 0f(x)limx i) + = +f(x)f(y) xy xf(y) yf(x) x, y y x ,: ( )( )+ = + + = = = =22 22f(x)f(x) x xf(x) xf(x) f(x) 2xf(x) x 0f(x) x 0 f(x) x 0 f(x) x f(x)=x . ii) = = +3 2x 0 x 0x 1lim limx x 6) (Oldies but goodies) i) 0x f(x) 0 g(x) 0 + =0x xlim(f(x) g(x)) 0 , = =0 0x x x xlim f(x) lim g(x) 0 ii)+ =02 2x xlim(f (x) g (x)) 0 = =0 0x x x xlim f(x) lim g(x) 0 iii) + + + 2 2f (x) g (x) 2f(x) 4g(x) 5 x , 0x xlim f(x) x 0lim g(x) . i) 0x : +0 f(x) f(x) g(x) + =0x xlim(f(x) g(x)) 0 , =0x xlim f(x) 0 . =0x xlim g(x) 0 . ii) + + + 2 2 2 2 2 2 2 2f (x) f (x) g (x) f (x) f (x) g (x) f(x) f (x) g (x) + +2 2 2 2( f (x) g (x)) f(x) f (x) g (x) + =02 2x xlim f (x) g (x) 0 =0x xlim f(x) 0 . =0x xlim g(x) 0 . iii) :

mathhmagic.blogspot.gr : ., ., ., ., . 7+ + + + + + + + 2 2 2 2f (x) g (x) 2f(x) 4g(x) 5 x f (x) 2f(x) 1 1 g (x) 4g(x) 4 4 5 x ( ) ( ) ( ) ( )+ + + + + + + 2 22 2f (x) 2f(x) 1 1 g (x) 4g(x) 4 4 5 x f(x) 1 g(x) 2 x ( ) ( ) + + 2 20 f(x) 1 g(x) 2 x , : = =x 0 x 0lim 0 lim x 0 ( ) ( )( ) + + =2 2x 0lim f(x) 1 g(x) 2 0 . (ii) : ( )+ =x 0lim f(x) 1 0 ( ) =x 0lim g(x) 2 0 = x 0lim f(x) 1 =x 0lim g(x) 2 . 7) ++5 3x 0 3 5x xlimx x =15 x u = 15x u 15=..(3,5) : ++ + + + += = = = = =+ + + ++ +5 1055 3 5 5 53 15 15 15 5 10 1053 3 3x 0 u 0 u 0 u 0 u 0 u 015 3 3 12 12 123 35 3 515 15u (u 1)x x u u u u u u 1 u 1lim lim lim lim lim lim 1u u u (u 1) u u 1 u 1x x u u 8) f, g R : ( ) ( ) ( ) ( )x 0 x 0g x x xf x g x x xf x 1lim 1 lim2x x 2 += = ( ) ( )x 0 x 0lim f x lim g x . : ( ) ( ) ( ) ( ) ( ) ( )g x x xf x g x x xf xh x x2x x += =, : ( ) ( )x 0 x 01lim h x =1 (1) lim x = (2).2 : ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x 0g x x xf x 2xh x (3)g x x xf x x x (4)_______________________________xh x x h x 12g x x 2xh x x x g x xxx 2 2x = + = + = + = + = + ( )( )( ) ( )(1),(2)x 0x 0 x 0 x 0x 0lim h x 1 1 1 1 5lim g x lim x lim g xx 2 1 2 2 4limx = + = + = (3): ( ) ( ) ( ) ( ) ( ) ( )(x 0) xxf x g x x 2xh x f x g x 2h xx = = , ( ) ( ) ( )x 0 x 0 x 0 x 0x 5lim f x lim g x lim 2lim h x 1 2 1x 4 = = : ( )x 03lim f x4= .

mathhmagic.blogspot.gr : ., ., ., ., . 89) 2x ( )x 1 1f(x)x+ + + = , 0 < < : ) f ) x 0lim f(x) i) : 2x ( )x 1 0 (1)+ + + x 0 : 2 2( ) 4 .. ( ) 0 = + = = > ( 0 < < ) : 21,2( ) ( ) 1( ) ( ) ( ) ( ) 2 x( ) ( ) 12 22 + + = + + = = = + = 0 < < , (1) : 1 10 . f : 1 1,0 0, . ) ( ) ( )( )2 22x 0 x 0 x 0 2x ( )x 1 1 x ( )x 1 1x ( )x 1 1lim f(x) lim limx x x ( )x 1 1 + + + + + + ++ + + = = =+ + + + ( )( ) ( ) ( )22 22 2x 0 x 0 x 02 2 2x ( )x 1 1x ( )x 1 1 x ( )xlim lim limx x ( )x 1 1 x x ( )x 1 1 x x ( )x 1 1 + + + + + + + + = = =+ + + + + + + + + + + + ( )x 0 x 0 2 22x(x ( )) x ( ) 0 ( ) lim lim2x ( )x 1 1 0 ( )0 1 1x x ( )x 1 1 + + + + + + += = = =+ + + + + + + ++ + + + 10) ++ 2xlim[ln( x 1 x)] + =2x 1 x u ( )( )+ + + ++ + + + + = = = =+ + + + + +2 22 22x x x x2 2 2x 1 x x 1 x x 1 x 1lim ( x 1 x) lim lim lim 0x 1 x x 1 x x 1 x ++ + = = 2x u 0lim[ln( x 1 x)] lim[ln(u)]

mathhmagic.blogspot.gr : ., ., ., ., . 911) 2 2f(x) 4x 4x 3 x 2x 2 3x 2= + + + + + + x 1 xx x 12 3 2g(x)2 3 2++ +=+ + i) xlim f(x)+ ii) xlim f(x) iii) xlim g(x)+ iv) xlim g(x) i) ( ) ( )2 2 2 2x x xlim f(x) lim 4x 4x 3 x 2x 2 3x 2 lim 4x 4x 3 2x x 2x 2 x 2+ + += + + + + + + = + + + + + + = ( ) ( )( )2 2 2 2x 2 2( 4x 4x 3 2) 4x 4x 3 2x x 2x 2 x x 2x 2 xlim 24x 4x 3 2x x 2x 2 x+ + + + + + + + + + + + + = + + + + + + 2 2 2 2x x2 2 2 24x 4x 3 4x x 2x 2 x 4x 3 2x 2lim 2 lim 24x 4x 3 2x x 2x 2 x 4x 4x 3 2x x 2x 2 x+ + + + + + + += + + = + + = + + + + + + + + + + + + x 0x x2 22 2x x2 24x 3 2x 2 4x 3 2x 2lim 2 lim 24 3 2 24x 4x 3 2x x 2x 2 xx 4 2 x 1x xx xx4x 3 2x 2lim 2 lim4 3 2 2x 4 2 x 1 1x xx x>+ ++ + + + + += + + = + + = + + + + + + + + + + + + + ++ + = + + + + + + 2 23 24 x 2x x24 3 2 2x 4 2 x 1 1x xx x + + + + = + + + + + + ( )( ) ( )x2 23 2 24 2 24 0x x x 4 2lim 2 2 2 44 24 3 2 2 4 0 0 2 1 0 0 14 2 1 1x xx x+ + + + + + + = + + = + + = + + + + + + + + + + + + ii) x 02 2x x2 2x4 3 2 2 2lim f(x) lim x 4 x 1 x 3x x xx x4 3 2 2 2lim x 4 x 1 x 3x x xx x> = + + + + + + = + + + + + = ( )( ) ( )2 2x4 3 2 2 2lim x 4 1 3x x xx x( ) 4 0 0 1 0 0 3 0 ( ) 6 + + + + + + + = + + + + + + + + = + = + iii) x 1 x x xx x 1 x xx x xxxxxxxxx xxxx2 3 2 2 2 3 2lim g(x) lim lim2 3 2 2 3 3 22 2 2 23 2 1 2 13 3 3 2 0 1 0 13lim lim0 3 0 32 22 233 33 33 3+++ + ++ + + += = =+ + + + + + + = = = + + + + + + iv) x 1 x x xx x 1 x xx x x2 3 2 2 2 3 2 2 0 0 2lim g(x) lim lim 10 3 0 22 3 2 2 3 3 2++ + + += = = =+ ++ + + +

mathhmagic.blogspot.gr : ., ., ., ., . 1012) f : +=x 0f(x) xlim 2x i) x 0lim f(x) x 0f(x)limx ii) , +=+2 2x 0xf(x) xxlim 5x x f (x) iii) ++x 0f(x) xlimx x i) =f(x) xg(x)x =x 0lim g(x) 2 = +f(x) xg(x) x , x 0lim g(x) : = + = + =x 0 x 0lim f(x) lim(xf(x) x) 0 2 0 0 + += = = + =x 0 x 0 x 0 x 0xg(x) x x(g(x) 1)f(x)lim lim lim lim(g(x) 1) 3x x x ii) ++ ++= = = =+ +++22:x 2 2 22 2 2 2 2 22x 0 x 0 x 0 x 0:x22 2 2xf(x) xx xxxf(x) xf(x)xf(x) xx x x x x xlim lim lim limx x f (x) x x f (x) x x f (x)f (x)xxx x x + += =+ + 2x 0xf(x)3 x xlim0 9f(x)xx., += =+3 5 ... 420 9 iii) ++++= = =+++ + x ,0 0,2 2x 0 x 0 x 0 x 0f(x)x1xf(x) x f(x)1f(x) x xx xlim lim lim limx x x xx x1 1x x x x 0f(x)limx =x u x 0 u 0 =u 0f(u)lim 3u. +++= = =+ ++x 0 x 0f(x)x1x 31f(x) x x 1lim lim 2xx x 1 11x 13) f,g , : ( ) ( )x xlim f(x) g(x) lim f(x) g(x) 0+ + = = x xlim f(x) lim g(x) 0+ += = x ( )( )( )( )( )( )22x xxx xlim f(x) g(x) 0 lim f(x) g(x) 0lim f(x) g(x) 4f(x) g(x) 0lim f(x) g(x) 0 lim 4f(x) g(x) 0+ +++ + = = + = = = ( )2xlim f(x) g(x) 0+ + = ( )( )2x xxlim f(x) g(x) 0 lim f(x) g(x) 0lim f(x) g(x) 0+ +++ = + = + =

mathhmagic.blogspot.gr : ., ., ., ., . 11 ( )( )( )( )x xx xx xx xlim f(x) g(x) 0 lim f(x) g(x) (f(x) g(x)) 0lim f(x) g(x) 0 lim f(x) g(x) (f(x) g(x)) 0lim 2f(x) 0 lim f(x) 0lim 2g(x) 0 lim g(x) 0+ ++ ++ ++ + = + + = + = + = = = = = 14) ( )+ f : 0, , + =3f (x) f(x) x , x : i) < x 0 ii) =x 0lim f(x) 0 iii) = +x 01limf(x) iv) +xlim f(x) , += +xlim f(x) i) + = + =3 2f (x) f(x) x f(x)(f (x) 1) x >x 0 =+2xf(x)f (x) 1 >x 0 . < x 0 ii) < x 0 : =x 0lim f(x) 0 iii) =x 0lim f(x) 0 >f(x) 0 0 = +x 01limf(x) iv) += xlim f(x) ( )+ + + = 3x xlim f(x) f(x) lim x ( )+ + = 3xlim f(x) f(x) += +xlim x += xlim f(x) L ( )+ + + = 3x xlim f(x) f(x) lim x ( )+ + = + 3 3xlim f(x) f(x) L L += +xlim x . 15)Bonus .( ) ) f : =x lim f(x) L = x lim f(x) L ) f : =x lim f(x) L =x lim f(x) L i)H f x x = f( x) f(x) (1) = = = = (1)x x x u lim f(x) lim f( x) lim f( x) lim f(u) L ii)H f x x =f( x) f(x) = = = =x ux x u lim f(x) lim f( x) lim f(u) L

mathhmagic.blogspot.gr : ., ., ., ., . 1216) ( ..1974) . i) + +2x 0x x 1974limxx ii) ( ) ( ) 2x 0 1974x 1975xlimx iii) + 2 2x 0(2x)lim 1974x2x, * iv) 2xxlimx v) +22x1974x xlimx x . x 0f(x)lim 1x= 22x 0f(13x 7x)lim13x 7x+ i) ( )+ + =2x 0lim x x 1974 1974 ( )=x 0lim xx 0 0. x 0,2 >x 0 , >x 0 >xx 0 x ,02 xx 0 , x 0. ( )=x 0lim xx 0 >xx 0 x 0 = + x 01limxx ( ) + += + + = = + = + 22x 0 x 0x x 1974 1lim lim x x 1974 .. ( ) 1974xx xx ii) ( ) ( ) ( ) ( ) ( ) ( ) = = = 2x 0 x 0 x 0 1974x 1975x 1974x 1975x 1974x 1975xlim lim lim 1974 1975x x 1974x 1975xx ( ) ( ) ( ) ( ) = = = u 1974xx 0 x 0 y 1975x u 0 y 0 1974x 1975x u ylim 1974 lim 1975 1974 lim 1975lim 1974 1 1975 11974x 1975x u y1974 1975 iii) . += = > + < 2 2x 0 2, 1974(2x)lim 1974x 2, 02x , 2,, iv) .

mathhmagic.blogspot.gr : ., ., ., ., . 13 = =2 2x x1 1lim 0 , lim 0x x (1) : =2xxlim 0x v) +++ += = = = 22 vi2 22 2x x x221974x x x19741974x x 1974 0x xlim lim lim 1974x 1 0x x x x1xx . ( )( )22 2 2 u 13x 72 2 2x 0 x 0 u 0 x 0 u 0 x 0x 13x 7f(13x 7) f(13x 7x) 13x 7x f(u) f(u) 13x 7lim lim lim lim lim lim 1( 1) 1u u 13x 7x 13x 713x 7 13x 7x 13x 7x= + ++ + + += = = = = + 17)oldies but goodies) : i) x1lim xx+ ii) x1lim xx+ iii) x1lim (x 1)x++ iv) 2x1lim ( 9x 1)x+ v) 2x1lim 2x 1 x+ vi) xx 4xlim2x x i) 1 1u xx u= = x x1lim u lim 0x+ += = , : u 0 u 0u1lim u lim 1u u = = x1lim x 1x= ii) (i)x x x x1 1 1 1 1lim x lim x lim lim x 0 1 0x x xx x+ + + += = = = iii) (i)x x x x1 1 1 1 1lim(x 1) lim x(1 ) lim(1 ) lim x (1 0)1 1x x x x x+ + + ++ = + = + = + = iv) 2 22x x1 1 1lim ( 9x 1) lim( x 9 )x xx + = + = x x 02 2x x1 1 1 1lim ( x 9 ) lim x 9 x xx x < + = + = 2x x1 1lim 9 lim x ( 9 0)1 3xx + = + = v) 1 1u xx u= = x x1lim u lim 0x+ += = ,

mathhmagic.blogspot.gr : ., ., ., ., . 14( )2 2x u 02 22 2 2u 0 u 0 u 02222u 0 u 01 1lim 2x 1 2 lim 1 ux uu u2 21 u 2 22 lim 2lim 2limu u u42u u 2 2lim lim 1 1uu22+ = = = = = = = = = vi) x x 0x xx x4xx 1xx 4xlim lim2 2x x x x4x 4xx 1 1x 1 0xlim lim 12 2 1x x x < = = + + = = = 18) f : , = f( ) 1-1 f(x) x x 1 xf (x) e 1 x : i) 1f (x) x x ii) =1 1x 0lim f (x) f (0) iii) =f(0) 0 iv) += +1xlim f (x) v) = x 0f(x)lim Lx =L 1 . i) f 1-1 1f = f( ) f(x) x x 1f (x) : 1 1 1f(f (x)) f (x) x f (x) = 1x f ( ) 1f (x) x ii) 1 xx f (x) e 1 (1) x =x 0 , =1 0 1 10 f (0) e 1 0 f (0) 0 f (0) 0 (1) =1x 0lim f (x) 0 =1 1x 0lim f (x) f (0) iii) = = =1 1f (0) 0 f(f (0)) f(0) 0 f(0) iv) 1 xx f (x) e 1 x ( ) = = +xx xlim x lim e 1 v) f(x) x x . , f(x) xx x x 0,2

mathhmagic.blogspot.gr : ., ., ., ., . 15+ + x 0 x 0f(x) xlim limx x L 1 (2) L 1 , f(x) xx x x ,02 x 0 x 0f(x) xlim limx x L 1 (3) (2) ,(3) =L 1 . 19)( G. Aligniac) I) f ( )0,+ : ( ) 2x x xe f x e e , ( )0,x + . lim ( ) 2xf x+= . ii) 2 *( ) 2 ,f x x x = . 61( )lim( 1)xf xx . . iii) f 4( ) 3 2 ( 1)xf x x x x x+ = + + , x . f(0). i) ( ) 2 ( ) 2 2 ( ) 2x x x x x x x x x x x xe f x e e e e f x e e e e e f x e e + 2 2( )2 ( ) 2 (1)x x x xx x xx x x x xe e e ee f x e ef xe e e e e + + ( *) lim 0xxxee+= lim 2 lim 2 2x xx xx xe ee e + + = + = , lim ( ) 2xf x+= . (*)(1 11 1xxx x xeee e e ( )0,x + 1 1lim lim 0x xx xe e+ += = lim 0xxxee+= ) ii) 61lim( 1) 0xx = , 61( )lim( 1)xf xx, ( ) ( )21 1lim ( ) 0 lim 2 0x xf x x x = = 1 2 0 1 = = . : ( ) ( ) ( )2 22 1 2 326 6 6 61 1 1 11 1 .. 1( ) 1 2 1lim lim lim lim( 1) ( 1) ( 1) ( 1)x x x xx x x x x xf x x xx x x x + + + + ++ += = = = ( ) ( )21 2 321 2 34 41 1.. 1 1lim lim .. 1 ( )( 1) ( 1)x xx x x xx x x xx x + + + + += + + + + + = + = + iii) 404 4 2 ( 1) 3( ) 3 2 ( 1) ( ) 2 ( 1) 3 ( )x x x x xxf x x x x x xf x x x x x f xx + + + = + + = + + = f 0

mathhmagic.blogspot.gr : ., ., ., ., . 164 4 40 0 0 0 0 02 ( 1) 3 2 ( 1) 3 2 (( 1) 3)(0) lim ( ) lim lim lim 2lim lim2x x x x x xx x x x x x x x x x xf f xx x x x x + + + + = = = + = + = ( )44 40 0 0 02 (( 1) 3) 22 lim lim 2lim lim ( 1) 3 2 1 (0 1) 3 02 2x x x xx x x xxx x x + + = + + = + + = 20) :f + =2( 2) ( 2) ( 2)lim 22xx f x xx. (0)f . : + = + = 2 2 2( 2) ( 2) ( 2) ( 2)lim 2 lim ( 2) lim 22 2x x xx f x x xf xx x (1) ( 2)f x 2x ( ) f . 2 : = = == =222 0lim ( 2) (2 2) 0 (2)( 2)lim lim 1 (3)2xu xx uf x fx ux u (2),(3) (1) ( ( ) + = +0 0 0lim ( ) ( ) lim ( ) lim ( )x x x x x xf x g x f x g x ) + =2 2( 2)lim ( 2) lim 22x xxf xx + = =(0) 1 2 (0) 1f f . 21) f =00x : =0( ) (0)lim 2xf x fx = 0( )limxf x xax ) Cf . ) . ) 0x : = = +( )( ) ( ) ( )f x xg x f x xg x xx ( ) = + = + =0 0lim ( ) lim ( ) 0 0 0x xf x xg x x a f =00x =0lim ( ) (0)xf x f . : =(0) 0f Cf . ) : = =0 0( ) (0) ( )lim 2 lim 2x xf x f f xx x : = = = = 0 0( ) ( )lim lim 2 1 1x xf x x f x xx x x 22)) f,g + =2 2 2( ) ( )f x g x x (1) x . f,g . ) f g (1) + =( ) ( )f x g x x x

mathhmagic.blogspot.gr : ., ., ., ., . 17 f g =00x . ) f g + =2 2( ) ( ) 2 ( )f x g x xf x x f g =00x . ) : + = 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )f x f x g x x f x x f x x x f x x (2) + = 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )g x f x g x x g x x g x x x g x x (3) x= (2) ,(3) (2) : =( ) 0 ( ) 0 ( ) 0f f f (3) : =( ) 0 ( ) 0 ( ) 0g g g ( ) = =lim 0,lim 0x xx x (2) ,(3) : = =lim ( ) 0 ( )xf x f ,= =lim ( ) 0 ( )xg x g . f,g . ) = 0x (1) =+ = + = =(0) 0(0) (0) 0 (0) (0) 0(0) 0ff g f gg (2) + = ( ) ( ) ( ) ( ) ( )f x f x g x x f x x x f x x ( ) ( ) = =0 0lim 0 , lim 0x xx x = =0lim ( ) 0 (0)xf x f f =00x . g =00x . ) : + = 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )f x f x g x x f x x f x x x f x x (2) + = 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )g x f x g x x g x x g x x x g x x (3) x= (2) ,(3) (2) : =( ) 0 ( ) 0 ( ) 0f f f (3) : =( ) 0 ( ) 0 ( ) 0g g g ( ) = =lim 0,lim 0x xx x (2) ,(3) : = =lim ( ) 0 ( )xf x f ,= =lim ( ) 0 ( )xg x g . f,g . ) = 0x : =+ = + = =2 2 2 2 (0) 0(0) (0) 2 0 (0) (0) (0) 0(0) 0ff g f f gg x ( )+ = + + = + =22 2 2 2 2 2 2 2( ) ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) ( )f x g x xf x f x g x xf x x x f x x g x x (*) (*) :

mathhmagic.blogspot.gr : ., ., ., ., . 18( ) ( ) ( ) + = 2 2 22 2 2( ) ( ) ( ) ( ) ( )f x x f x x g x x f x x x f x x x + +( ) ( )x f x x x x x f x x x ( ) ( ) + = + =0 0lim 0 , lim 0x xx x x x ( )=0lim ( ) 0xf x f =00x . g =00x . 23) :f x : ( 2) ( ) ( 2) 1x f x x (1) . (2)f . f 2. =2lim ( ) (2)xf x f (2). 2lim ( )xf x . 2x . > >2 0 2x x (1) : = + 2 2( 2) 1 ( 2) 1( ) lim ( ) lim( 2) ( 2)x xx xf x f xx x (3) = 2u x + 2x + 0u (3) : 2 0 21lim ( ) lim lim ( ) 0x u xuf x f xu (4) <

mathhmagic.blogspot.gr : ., ., ., ., . 19: ( ) ( ) ( ) ( ) + + + + = = = + = 2 2 2 2 2 22 2 2 2444 1 444 1 1 1 1 1 1( ) 444x x x x x xf xx x x x ( ) ( )( )( )( )( )( ) + + + + + = + = + = + = + + + + + + 22 22 2 22 2222 2 2 2 2 21 11 1 1 1 1 1444 444 4441 1 1 1 1 1xx x xx x xx xxx x x x x x ( )( )( )( ) + = + = + = + + ++ + + + 2 22 2 222 2 2 21 1 1444 444 4441 11 1 1 1x xx x xx x xxx x x x = + = + = + = = + + + + 2 20 0 0 02 21 1 1 3lim ( ) lim 444 444 lim lim 444 1 444 6662 21 1 1 1x x x xx xf xx xx x =(0) 666f . : ( ) + = =2 22444 1, 0( )666 , 0x xxf xxx ) : ( )= + 2 2 2( ) 444 1x f x x x x ( ),0 . : ( ) ( ) = + = + = + = 22 2 2 2 22 21 1lim ( ) lim 444 1 lim 444 1 lim 444 1x x x xxx f x x x x x x xxx x( ) > = + = + + 2 202 21 1lim 444 1 lim 444 1xx xx xx x xx xx x (2) : 2 21 1 1x xx xx x x = = 1 1lim 0 , lim 0x xx x =2lim 0xxx (3) (2) ,(3) ( ) ( )( )( ) ( )( ) = + + + = + = + 2lim ( ) 444 ( ) 0 1 0 444 ( ) 1x x f x

mathhmagic.blogspot.gr : ., ., ., ., . 2025) :f 2 ( ) ( ( ))f x f f x x = , x . .i) f 1-1. ii) 22 ( 1974 ) ( (0))f x x f f = . ( )lim , 0xf xx += > : i) lim ( )xf x+ ii) 1 = i) : 2 ( ) ( ( ))f x f f x x = (1) 1 2,x x , 1 2( ) ( )f x f x= : ( )1 2 1 21 21 2 1 21 1 2 2 1 2( ( )) ( ( )) ( ( )) ( ( ))( ) ( )2 ( ) 2 ( ) 2 ( ) 2 ( )2 ( ) ( ( )) 2 ( ) ( ( ))f f x f f x f f x f f xf x f xf x f x f x f xf x f f x f x f f x x x+ = = = = = = = , f 1-1. ii) (1) 0x = 2 (0) ( (0)) 0 2 (0) ( (0))f f f f f f = = , (2)2 21 12 2( (0)) 2 ( 1974 ) 2 (0) 2 ( 1974 )(0) ( 1974 ) 1974 0 0 1974ff f f x x f f x xf f x x x x x x= = = = = = . ( )( ) ( ) ( )f xg x xg x f xx= = , 0x > + ( )lim ( )0lim ( ) lim ( ) ( )xg xx xf x xg x+=+ + >= = + = + ii) lim ( )xf x+= + >0 ( ),x + ( ) 0f x > . ( ),x + : 0 ( ( )) 2 ( ) ( ( ) 2 ( )2 ( ) ( ( )) ( ( )) 2 ( ) 1x f f x x f x f f x f xf x f f x x f f x x f xx x x x> + = + = = + = ( ( ) 2 ( ) ( ( )) ( ) 2 ( )1 1 (3)( )f f x f x f f x f x f xx x f x x x+ = + = (*)( ( )) ( ) 2 ( ) ( ( )) ( ) 2 ( )lim 1 lim lim lim 1 lim( ) ( )x x x x xf f x f x f x f f x f x f xf x x x f x x x+ + + + + + = + = 21 2 2 1 0 1 + = + = = (*) ( )u f x= lim ( )xf x+= + x + u + ( )( ( )) ( )lim lim( )u f xx uf f x f uf x u=+ + = =

mathhmagic.blogspot.gr : ., ., ., ., . 2126) f: : (1) ( ) ( ) ( )f x f y x y x y + ,x y , i) ( )f x x x = + x . ii) 0( ( ))limxf f xx. i) (1) y x x y ( ) ( ) ( ) ( ) ( ) ( )f y f x y x y x f x f y x y x y + + (2) (1) ,(2): ( ) ( ) ( )f x f y x y x y = + (3) ,x y , H Cf (0) 0f = (3) y x ( ) (0) 0 ( 0) ( )f x f x x f x x x = + = + ii) ( )(3),0 0 0 0( ( ))( ( )) ( ) ( )lim lim lim lim 1y xx x x xx x x xf f x f x x x x xx x x x x = + + += = = + + = 0 0 00( ) (1 )( )lim 1 lim 1 lim 1(1 ) (1 )lim 1 1 11 1x x xxx x x x x xxx xx x x x xx x x x x xx xx x x x x xxxx xx x + + ++ + = + + = + + = + + + + = + + = ( )22 1 + + + = + (ii) ( )u f x= ( )0 0lim ( ) lim 0 0 0x xf x x x = + = + = 0u 0x : 0 0 0 0( ( )) ( ( )) ( ) ( ( )) ( )lim lim lim lim( ) ( )x x x xf f x f f x f x f f x f xx f x x f x x = = (4) 0 0 0 0( ( )) ( )lim lim lim lim 1( )x u u uf f x f u u u uf x u u u + = = = + = + 0 0( )lim lim 1x xf x x xx x += = + (4) : ( )20 0 0( ( )) ( ( )) ( )lim lim lim 1( )x x xf f x f f x f xx f x x = = + 27) :f : + = + +( ) ( ) ( ) 1974f x y f x f y xy (1) ,x y . ) (0)f ) f 0 x . ) (1) = = 0x y : + = + + = =(0 0) (0) (0) 1974 0 0 (0) 2 (0) (0) 0f f f f f f ) f 0 = =0 0lim ( ) (0) lim ( ) 0x xf x f f x (2) *0x 0x f =0x x h 0x x 0h = + = + + = + + 00 0 0 0 00 0 0 0 0lim ( ) lim ( ) lim ( ) ( ) 1974 lim ( ) lim ( ) lim 1974x x h h h h hf x f x h f x f h x h f x f h x h

mathhmagic.blogspot.gr : ., ., ., ., . 22= + + =0 0( ) (0) 0 ( )f x f f x f *0x x . 28) = 1( ) 1xf x e = + +( ) ( 1) 1g x ln x ) =( )f x x ( )1,3 . ) ( )1,3 lim ( )xg x . ) = ( ) ( )h x f x x (1) h 1,3 . ( )( ) ( ) = =

mathhmagic.blogspot.gr : ., ., ., ., . 2331)( 1995) ) f ( ) 0f x x . f . ) ( )+ : 0,f > 1x : = + +2( ) 1 11f x xx. f ( )+1, . ) f . , <

mathhmagic.blogspot.gr : ., ., ., ., . 2433) :f 2 2(1) (4) 2 (1) 6 (4) 10f f f f+ = + 2 ( ) 4 ( ) 3 0f x f x + , x ) (1)f (4)f . ) f m . ) f . ) f , . ) : 2 2 2 2(1) (4) 2 (1) 6 (4) 10 (1) 2 (1) 1 (4) 6 (4) 9 0f f f f f f f f+ = + + + + = ( ) ( )2 2(1) 1 0 (1) 1(1) 1 (4) 3 0(4) 3 0 (4) 3f ff ff f = = + = = = ) 2 ( ) 4 ( ) 3 0 ... 1 ( ) 3f x f x f x + x . (1) 1f = (4) 3f = (1) ( ) (4)f f x f x . , f 1 (1)m f= = 3 (4)M f= = . ) 1 4< (1) (4)f f< 5 4> (5) (4)f f f . ) f (1) 1f = (4) 3f = . ( ) 1,3f A = . 34) Bolzano) f: ( ( ))f f = . ( )f x x= . : ( )f = x=. ( )f < ( ) ( )g x f x x= , , ( )f . ( ) ( )g f = ( ( )) ( ( )) ( ) ( )g f f f f f = = ( )( ) ( )2( ( )) ( ) ( ) ( ) ( ) 0g f g f f f = = < Bolzano ( )0 , ( )x f 0 0 0 0 0( ) 0 ( ) 0 ( )g x f x x f x x= = = ( )f > ,

mathhmagic.blogspot.gr : ., ., ., ., . 2535) f : + =21( ) 2 ( 1)lim 1001xf x xx i) Cf (1,2). ii) : 1( ) 2lim1xf xx 1( ) 3 1lim1xf xx iii) ( )f x : = =1 23 3,2 2x x : 2lim ( ) 0xf x , ( )f x . i) ( ) ( ) + = = + = + 22 2( ) 2 ( 1)( ) ( ) 1 ( ) 2 ( 1) ( ) ( ) 1 2 ( 1)1f x xg x g x x f x x f x g x x xx : ( )( ) ( ) ( ) ( ) = + = + = + =2 2 21 1 1 1 1lim ( ) lim ( ) 1 2 ( 1) lim ( ) lim 1 lim 2 ( 1) 100 0 2 0 2x x x x xf x g x x x g x x x (1) f =01x = =(1)1lim ( ) (1) 2 (1)xf x f f Cf (1,2). ii) ( ) + = = + = + = 2 2( ) 2 ( 1) ( ) 2 ( ) 2 ( ) 2( 1)( ) ( ) ( ) 1 ( ) 11 1 1 1 1f x x f x f x f xxg x g x g x x g x xx x x x x ( )( ) = = = 1 1 1 1( ) 2 ( ) 2 ( ) 2lim ( ) 1 lim 100 (1 1) lim lim 1001 1 1x x x xf x f x f xg x xx x x = < (2) 0f ( )f x - + +

mathhmagic.blogspot.gr : ., ., ., ., . 2636) f [ ]0,1 ( )0,1 . ( ) ( )f 0 2 f 1 4= = , : . y 3= f ( )0x 0,1 . . ( )0x 0,1 , ( )01 2 3 4f f f f5 5 5 5f x4 + + + = (1). . ( )y f x= y 3= ( )0x 0,1 , ( )0f x 3= . g ( ) ( )g x f x 3= (2), [ ]0,1 . i) g [ ]0,1 . ii) ( ) ( ) ( )[ ] ( )[ ] ( ) ( )(2)g 0 g 1 f 0 3 f 1 3 2 3 4 3 1 1 1 0 = = = = < . Bolzano ( )0x 0,1 ( ) ( ) ( )(2)0 0 0g x 0 f x 3 0 f x 3= = = = , fC y 3= ( )0,1 . g . f ( )0,1 ( )1 2x ,x 0,1 1 2x x< ( ) ( ) ( ) ( ) ( ) ( )(2)1 2 1 2 1 2f x f x f x 3 f x g x g x< < < , g ( )0,1 0x . ( )0x 0,1 y 3= f . . f [ ]0,1 m . : ( )m f x M< < (3), [ ]x 0,1 . 1 2 3 4, , ,5 5 5 5 , (3): 1m f M52m f M53m f M54m f M5 : 1 2 3 44m f f f f 4M5 5 5 51 2 3 4f f f f5 5 5 5m M (4)4 + + + + + + f ( )0,1 m M . (4) ( )0x 0,1 : ( )01 2 3 4f f f f5 5 5 5f x4 + + + = .

mathhmagic.blogspot.gr : ., ., ., ., . 2737) f 0,1 =(0) (1)f f . g : = +1( ) ( ) ( )g x f x f x , * . g. B. : i) > 1, + + + + =1 2 1(0) ( ) ( ) ... ( ) 0g g g g ii) : = +1( ) ( )f x f x )0,1 . ( ) ) + 0 11010 1gxx D xx = 10,gD .i) 10,x ,= +1( ) ( ) ( )g x f x f x = 1(0) (0) ( )g f f = 1 1 2( ) ( ) ( )g f f = 2 2 3( ) ( ) ( )g f f = 1 1( ) ( ) (1)g f f : + + + + = =1 2 1(0) ( ) ( ) ... ( ) (0) (1) 0g g g g f f ii) f 10, : 10,x , + ( )(0)1( )( ).....................1( )m g Mm g Mm f x Mm g M + + + + 1 2 1(0) ( ) ( ) ... ( ) 0 0m g g g g m m 010,x =0( ) 0g x ( , f ( )3,3 , ( ) 0f x > ( )3,3x : ( ) 9f x x= , ( )3,3x 2 2( , ) 9M x y Cf x y + = , 0y . (0,0) 3 = .

mathhmagic.blogspot.gr : ., ., ., ., . 2940) ( 1974 ;;;;) ( )+ : 0,f ( )0,+ , =2 2( ) lnf x x > 0x i) =( ) 0f x ii) f (0,1) +(1, ) . iii) (1974) 0f . . i) = = = = =2 2( ) 0 ( ) 0 ln 0 ln 0 1f x f x x x x =( ) 0f x 1 .ii) f (0,1) +(1, ) . , f . iii) f ( )0,1 : ln 0x ( )+1, = =(1) 0 ln1f =( ) lnf x x > 0x . 41) :f : ( ) 0f x 0x + =(1974) ( 1974) 0f f i)

mathhmagic.blogspot.gr : ., ., ., ., . 3042) : 0,f : (1) 2 2( ) 1f x x+ = 0,x i) f ( )0, . ii) f ( )1 0,x 1( ) 8f x = . iii) ( ) 3121 2( ) 2 4 1974lim( ) ( )xf x x xf x f x x++ + ( )2 0,x i). f ( )0, ( )0, ( ) 0f x ( )0,x . ( )0 0,x 0( ) 0f x = . (1) 0x x= 2 2 20 0 0 0 0( ) 1 1 0f x x x x x + = = = = , ( )0 0,x ( ) 0f x ( )0,x . f ( )0, . ii) f 1( ) 8f x = ( ) 0f x > ( )0,x . iii) 1( ) 8f x = : ( ) ( )23 3 312 2 21 2 2 2( ) 03 32 222 2( ) 2 4 1974 8 2 4 1974 10 4 1974lim lim lim( ) ( ) 8 ( ) 8 ( )10 5 5lim lim lim4 ( )8 ( ) 4 ( )x x xf xx x xf x x x x x x xf x f x x f x x f x xx x xf xf x x f x x+ + +>+ + ++ + + + += = == = = = + 43). f ( ) ( )31f x x x2= + . ). f . )). 1f . ). ( )1f x 64 = . . f R ( )f 8 6= x R ( ) ( )( )f x f f x 2= . ( )f 2 . . ). f [ )0,+ . 1 1 . ( )1 2, 0,x x + 1 2x x< . 3 31 2 1 2x x x x< < . : 3 31 1 2 2x x x x+ < + . ( ) ( )3 31 1 2 21 1x x x x2 2+ < + . ( ) ( )1 2f x f x< , f , . ). f [ )0,+ . f : ( ) ( ))xf 0 , lim f x+. : ( ) ( )31f 0 0 0 02= + = ( ) ( )3x x1lim f x lim x x2+ += + = + . f [ )0,+ . f 1f , [ )1f : 0, R + .

mathhmagic.blogspot.gr : ., ., ., ., . 31). [ )x 0, + , ( ) ( )( ) ( ) ( ) ( ) ( )1 1 31 1f x 64 f f x f 64 x f 64 64 64 8 4 x 62 2 = = = = + = + = . . ( ) ( )( )f x f f x 2= x R x 8= . ( ) ( )( )f 8 f f 8 2= ( )f 8 6= , ( ) ( )16 f 6 2 f 63 = = . ( ) ( )f 6 ,f 8 f [ ]6,8 ( ) ( )0 0x 6,8 f x 2 = (1). 0x x= ( ) ( )( )0 0f x f f x 2= (1) ( ) ( )2 f 2 2 f 2 1 = = . 44))(oldies but goodies) f =( )A f A , : = =1( ) ( ) ( )f x f x f x x ( )x A f A ) = ( ) ln , 1f x x x x 1,Cf Cf ) f , 1-1, . ( )x A f A =( )f x x . : = 1( )x f x = 1( ) ( )f x f x ( )x A f A = 1( ) ( )f x f x . ( )f x x . ( )f x x =( )f x x ( )x A f A . ) = lny x =y x ) +1,x . f . f ) +0,x , : =(1) 0f ( )+ += = +lim ( ) lim lnx xf x x x ) = +( ) 1,A f A . f 1-1, , . 1f ) +1, ( 1f ) . 1,Cf Cf =y x , (). : )= = + = = =11( ) ( ) ( ) , 1, ln (ln 1) 0xf x f x f x x x x x x x x x e 1,Cf Cf ) +1, ( , )A e e

mathhmagic.blogspot.gr : ., ., ., ., . 3245) = >ln , 0xe x ) ( 0,1 ) = ( ) lnxf x e x ( )+1, i) f ( )+1, . ii) . iii) . ) < 0 1x ln 0x > 0xe ln 0xe x =lnxe x > 0 ( 0,1 . )i) xe ln x ( )+1, ( > 1x > >ln ln1 ln 0x x ) ( ) +1 2, 1,x x

mathhmagic.blogspot.gr : ., ., ., ., . 33( )( )( )( ) ( )( )( ) ( ) ( )(1),(2)2 2 2 2 2 22( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 2 4 0f g f g = + + + + + + == + + + + + + = + + =+ + = + Bolzano . 47).( ..) : 2015,2016f . ( )0 2015,2016x : 03 ( ) (2015) (2016) (2017)f x f f f= + + f 2015,2016 2015 2016 2017< < , : (2017) (2016) (2015)(2017) (2015) (2015)(2017) (2017) (2015)3 (2017) (2016) (2015) (2017) 3 (2015)f f ff f ff f ff f f f f< 0 2 ( ) ( )f x f x x+ = 2 ( ) ( ) 12 2 2f x f xx x+ = ( ) ( )12 2f x f xx x = ( )( ) ( )1 12 2 2 2f xf x f xx x x x = = ( ) 1 12 2f xx x ( )1 1 12 2 2f xx x x ( )1 1 1 12 2 2 2f xx x x + + 1 1 1 1 1lim lim2 2 2 2 2x xx x+ + + = + = : ( ) 1lim2xf xx+= 50) : ( )f : 0, R+ ( ) xf x e ln x x 1= + + . . : ( ) ( )xx 0lim f x lim f x+ . . x R ( )f x = . . ( )f x e= . . R : ( ) ( )2 1 2 2 2e e ln 2 ln 1 2 1 + = + + . . x 0x 0 x 0lim e e 1, lim ln x+ + = = = . ( )x 0lim x 1 1+ = , ( )x 0lim f x+= . : ( )xx x xlim e lim ln x lim x 1+ + = + = = . : ( )xlim f x= +

mathhmagic.blogspot.gr : ., ., ., ., . 35. ( ) ( )xx 0lim f x , lim f x+ = = + f , , + R. ( )f x = R. . xe ,ln x,x 1 ( )0,+ f ( )0,+ ( )f x = . . ( ) xf x e e ln x x 1 e= + + = . x 1= ( 1e ln1 e e+ + = ) . . . : ( ) ( ) ( ) ( )( )2 21 2 2 2 1 2 22e ln 1 e ln 2 2 1 e ln 1 1 1e ln 2 2 1 + ++ + + = + + + + + + == + + ( ) ( )2f 1 f 2 + = (1). f 1 1 (1): ( )22 21 2 2 1 0 1 0 1 + = + = = = . (*)Bonus (. ) f: 1-1 . 1 2 3, ,x x x < 1 2 3( ) ( ) ( )f x f x f x 2( )f x 1 3( ), ( )f x f x . : < >1 3 2( ) ( ) ( )f x f x f x (2) < >2 1 3( ) ( ) ( )f x f x f x (4) (1) . 3( )f x 1 2( ), ( )f x f x f , : ( ) =1 2 3( , ) : ( )x x f f x

mathhmagic.blogspot.gr : ., ., ., ., . 36, : f(x) x g(x) + + f() g() g(x) g()f(x) f() x f(x) x g(x) f(x) f() x g(x) g()x x x = + + + g(x) g()f(x) f()1x x + x x g(x) g()f(x) f()lim 1 limx x + , f ,g , f '() 1 g '() g '() f '() 1+ (1) x < : f(x) x g(x) + + f() g() x g(x) g()f(x) f() x f(x) x g(x) f(x) f() x g(x) g()x x x = < + + + g(x) g()f(x) f()1x x + x x g(x) g()f(x) f()lim 1 limx x + , f ,g , f '() 1 g '() g '() f '() 1+ (2) (1) (2) : g '() f '() 1 =

mathhmagic.blogspot.gr : ., ., ., ., . 4413) x 13f(x) 2 ln x 2 ln13 ,13 x= + i) f . ii) 2 169ln13 26 13 . iii) 2x x 169ln13 26x= i) fx D (0, ) = + 2 22 2 2x 13 2 1 13 26x x 169 (x 13)f '(x) 2 ln x 2 ln13 '13 x x 13 x 13x 13x = + = = = f '(x) 0< x 13 f x=13 f . ii) ( )( )f 0, 13 13 13 13 13 f() f(13) 2 ln 2 ln13 2 ln13 2 ln13 2 ln ln13 013 13 13 13 + + + + 2 2 2 2 2 13 13 1692 ln ln ln13 13 13 26 13 26 iii) ( )2 2 2 2 2 2x x 13 x x 13 x 13 x 13ln 2l n 2 ln x ln13 2 ln x 2 ln1313 26x 13 13x 13x 13x 13 xx 132 ln x 2 ln13 0 f(x) 013 x = = = = + = = f(x) 0= x=13 . 14)( 2000) f : : 2xx 0f(x) e 1lim 02x += i) f(0) . ii) f 0x 0= iii) xh(x) e f(x)= , Cf Ch A(0,f(0)) B(0,h(0)) . i) 2xf(x) e 1g(x)2x += , 0. : 2x2x 2xf(x) e 1g(x) g(x)2x f(x) e 1 f(x) g(x)2x e 12x += = + = + ( )2xx 0 x 0lim f(x) lim g(x)2x e 1 0 = + = , f x=0 x 0f(0) lim f(x) 0= = . ii) 2x 2x 2xx 0 x 0 x 0 x 0g(x)2x e 1 g(x)2x 2xf(x) f(0) e 1 e 1lim lim lim lim 2g(x)x 0 x x x 2x x + = = + = + =

mathhmagic.blogspot.gr : ., ., ., ., . 45( )2x 2x2 0x 0 x 0 x 0 x 02x 2xe 1 e 1lim 2g(x) 2 lim g(x) lim lim 2 0 1 2e 22x x 2x x = + = + = + = : . 2x 2x 2 02xx 0 x 0e 1 e elim lim k'(0), k(x) ex x 0 = = = f '(0) 0= iii) x 0 xx xx 0 x 0 x 0 x 0 x 0 x 0e f(x) e f(0) e f(x) f(x) f(x) f(0)h '(0) lim lim lim e lim lim e lim f '(0)x 0 x x x 0 = = = = = Cf Ch A(0,f(0)) B(0,h(0)) . 15) f : f '(x) 0 x h f(x)h(x) ,xf '(x)= . () Ch x'x () 4. 0 0(x ,h(x )) Ch x'x , 00 00f(x )h(x ) 0 0 f(x ) 0f '(x )= = = (1) f '(x) 0 0x . () 0 0(x ,h(x )) 4 0h'(x ) 14= = . : ( )( )( )22 2f '(x) f(x)f ''(x)f(x) f '(x)f '(x) f(x)f '(x)h '(x) 'f '(x) f '(x) f '(x) = = = , ( )( )( )( )2 2(1)0 0 0 00 2 20 0f '(x ) f(x )f ''(x ) f '(x )h '(x ) 1f '(x ) f '(x )= = = () 4.

mathhmagic.blogspot.gr : ., ., ., ., . 4616) f,g : : xg(x)f(e ) f(3x) f(e )2+= (1) f 1-1 : i) xe 3x= ( )0,1 . ii) g() = . iii) f ,g f(x) 0 x : 1g '()2+= iv) () g (,f()) x,Oy : 1()2 < i) xh(x) e 3x= , 0,1 0 1h(0) e 3 0 1 0 , h(1) e 3 1 e 3 0= = > = = < Bolzano ( ) 0,1 h() 0 ...e 3= = ii) (1) x= : 3 e 3g() g() f 1 1g() g() g() f(e ) f(3) f(e ) f(e )f(e ) f(e )2 22f(e )f(e ) f(e ) f(e ) e e g() 2=+ += = = = = = ii) (1) ( ) ( )xg(x) g(x) g(x)x xf(e ) f(3x) 1' f(e ) ' f '(e )e 3f '(3x) f '(e )e g '(x)2 2 += + = x=: ( ) ( )g() g() g() e 1 1f '(e )e 3f '(3) f '(e )e g '() f '(e )e 3f '(e ) f '(e )e g '()2 2==+ = + = f '(e ) 0 f '(e )e 3f '(e ) 2f '(e )e g '() 0 f '(e )(e 3 2e g '()) 0+ = + = e 3 1e 3 2e g '() 0 3 3 2 3g '() 0 g '()2= ++ = + = = iii) Cg (,g()) : 1 1 1 1 1y g() g '()(x ) y (x ) y x y x2 2 2 2 2+ + + + = = = + = + Cg x,Oy 2 1 B(0, ), ( ,0)2 1 + ( )< < = = =+ + +22 20 1 1 1 1 1 1 ()2 2 1 2 2 1 4( 1). ( ) ( ) ( ) ( ) ( )2 20 12 2 2 1 1 11 1 2( 1) 1 2 2 2 1 2 2 24( 1) 2 2( 1)<

mathhmagic.blogspot.gr : ., ., ., ., . 47 2 3 3 2 2 2 2 2 2 0 + < + < 0 1< < 17)i) f : g : : f(x)g(x) x= , x f g . ii) f : f '(x) 0 , x (3) 24 2f(x ) f(x ) = x (4) f A(1,1) . i) x 0= : f(0)g(0) 0 f(0) 0 g(0) 0= = = f g . Cf Cg (0,0) : f(0) 0= g(0) 0= (2) f(x)g(x) x= : ( )f(x)g(x) ' x ' f '(x)g(x)' f(x)g '(x) 1= + = x 0= f '(0)g(0)' f(0)g '(0) 1 0 1+ = = . ii) : f(1) 1= (4) : ( ) ( ) ( ) ( )( )24 2 4 4 2 23 4 2 2 2 3 4 2 2f(x ) ' f(x ) ' x ' f '(x ) 2 f(x ) ' f(x )4x f '(x ) 2f '(x ) x ' f(x ) 4x f '(x ) 4xf '(x )f(x ) = = = = x 1= :f '(1) 03 4 2 24 1 f '(1 ) 4 1f '(1 )f(1 ) f '(1) f '(1)f(1) f '(1)(1 f(1)) 0 f(1) 1 = = = = A(1,1) Cf . 18) f(1) 4,g(1) 8,f '(1) 4,g '(1) 10= = = = , : ) (f g)'(1) 51 = ) (f g)'(1) 72 = ) (f g)'(1) 14+ = ) f( )'(1) 40g= () (f g)'(1) f '(1) g '(1) .. 14+ = + = = () (f g)'(1) f '(1) g(1) f(1) g '(1) 32 40 72 = + = + =

mathhmagic.blogspot.gr : ., ., ., ., . 48 (18) , , , . : ( )2f(x) x ,x ,1 2,= + 1 f '(1) 2= )g(x) 3x,x 1,= + 1 g '(1) 3= f+g )2, {1} + 1. 19) f : f '(x) x . f(x)g(x)f '(x)= A(,) Cg xx. i) g . ii) , Cg . i) Cg xx 0= f()g() 0 0 f() 0f '()= = = f() 0x x x x x x f(x) f() f(x)g(x) g() f(x) 1 f(x) 0 1 f(x) f() 1f '(x) f '() f '(x)lim lim lim lim lim limx x x x f '(x) x f '(x) x f '(x)= = = = = = = x x x f(x) f() 1 f(x) f() 1 1lim lim lim f '() 1x f '(x) x f '(x) f '() = = = = g '() 1= g . i) 0g'() 1 45= = = 0 45= 20) ( )f : 1,+ f(x) x f(x)x e = . Cf (): 1y x 54= + Cf . . 1() : y x 54= + 14= . 0 0(x ,y ) Cf () : 0 1f '(x ) 4= = (1) f(x) . : f(x) x f(x) f(x) x f(x)x e ln(x ) ln(e ) f(x) ln x x f(x) f(x)ln x f(x) xf(x)(ln x 1) x = = = + = + = f x 1 ln x 0 ln x 1 1> > + > ln x 1 0+ . xf(x)ln x 1=+ (2) f

mathhmagic.blogspot.gr : ., ., ., ., . 49( ) ( )( ) ( ) ( )2 2 21ln x 1 x(x)' ln x 1 (x) ln x 1 'x ln xxf '(x) 'ln x 1 ln x 1 ln x 1 ln x 1+ + + = = = = + + + + (3) H (1) (3) : ( )( ) ( ) ( )2 2 20 0 0 0 0 0 0 020ln x 14 ln x ln x 1 4 ln x ln x 2 ln x 1 ln x 2 ln x 1 04ln x 1= = + = + + + =+ ( )20 0 0ln x 1 0 ln x 1 x e = = = e e ef(e)ln e 1 1 1 2= = =+ + eM e,2 (). : y f(e) f '(e)(x e) = , ( ) ( )2 2ln e 1 1f '(e)4ln e 1 1 1= = =+ + e 1 1 ey (x e) ... y x2 4 4 4 = = + 21) *f : 21 1f(x)xx= M(,f()), 0 f. ) Cf . )i) 3= Cf. ii) 1= Cf. ) 2 2 2 21 1 1 x 1 xf(x)xx x x x= = = ( ) ( )( ) ( )2 2 22 4 42 2 24 4 4 31 x 'x 1 x x ' x 1 x 2x1 xf '(x) 'x x xx 2x 2x x 2x x(x 2) x 2x x x x = = = = + = = = = Cf : 2 3 32 3 2 2 31 2 2y f() f '()(x ) y x 1 2 2 1 2 2y x y x = = + = + = + 2 3 3 23 2 2 2 3 2y x y x = + = + (2) )i) 3= M(3,f(3)) f 2M(3, )9 3 23 2 3 2 3 1 1y x y x27 33 3 = + = Cf 22222 21 x 1 11 1x 3 x 3(x 3) 0xy x27 327 3 x .. 1 3 21 x1 x 1 x y yyy y 93xx x = = == = = = = = =

mathhmagic.blogspot.gr : ., ., ., ., . 50 =3 2M(3, )9 Cf. ii) =1 M(1,f(1)) f M(1,0) 3 21 2 1 2 1y x y x 11 1 = + = + Cf 2y x 1x 1 x 1...1 xy 0 y 2yx = + = = = == , (1,f(1)) (1,0) (-1,2). 22) , , , , . 1) f Cf . 2) f , ( ), ( )0x , : 0f '(x ) 0= f() f()= . 3) 0x 0x . 4) , . 5) f 0x g 0x , f g 0x . 6) 0f '(x ) 0= f 0x . 7) f , : f '(x) 0,< x . 1) 2) 3) 4) 5) 6) 7) 23) f 3 2f(x) x x 2x 2016,,= + + + : 1 22x x3 = x x O 1x 2x fC

mathhmagic.blogspot.gr : ., ., ., ., . 51 1 2x ,x .Fermat 1 2f '(x ) 0,f '(x ) 0= = f : ( )3 2 2f '(x) x x 2x 2016 ' 3x x 2= + + + = + + 1 2x ,x 23x x 2 0+ + = Vieta (P= ) :1 22x x3 = 24) f : (1) 2 82f(x ) xf(x 1) x 2 0+ + + = x x 1g(x) f(x) e 3,x= + + . g A(1,g(1)) . i) f(1) 1,f '(1) 1= = ii) , f B(0,f(0)) ( 10)x y 1974 + = i) (1) x 0= : 2 82f(0 ) 0f(0 1) 0 2 0 2f(0) 2 0 f(0) 1+ + + = + = = (1) x 1= : 2 82f(1 ) 1f(1 1) 1 2 0 2f(1) 1 1 2 0 f(1) 1+ + + = + + = = Cg A(1,g(1)) xx, g'(1) 0= . x 1g'(x) f '(x) e = + 1 1g'(1) f '(1) e 0 f '(1) 1= + = = . ii) Cf B(0,f(0)) () : ( 10)x y 1974 + = f '(0) 10 = (1) : ( )2 2 7 2 72f '(x ) x ' x' f(x 1) xf '(x 1) 8x 0 4xf '(x ) f(x 1) xf '(x 1) 8x 0+ + + = + + + = x 1= : + + + = + + + = + + = = + = 2 74 1 f '(1 ) f(1 1) 1f '(1 1) 8 1 0 4f '(1) f(0) f '(0) 8 04 ( 1) 1 f '(0) 8 0 f '(0) 4 1 8 3 f '(0) 10 3 10 13= = = . 25)i) x 2e (x 1) 4 = . ii) xf(x) e= 1g(x)x= . Cf Cg. i) x 2h(x) e (x 1) 4= , ( )1 2 3 3h(1) e (1 1) 4 4 0,h(3) 4e 4 4 e 1 0= = < = = > Bolzano ( ) 1,3 h() 0= , x 2e (x 1) 4 = . ii) f,g . 0 0(x , y ) 0 0 0f(x ) g(x ) y= = 0 0f '(x ) g '(x )= , :

mathhmagic.blogspot.gr : ., ., ., ., . 52 0x01ex= 0x201ex= 0x 02 20 0 0 0 0 0 02001 1x x x x 0 x (1 x ) 0 x 1xx= = + = + = = , 11e . A(,f()) , (,g()) f g . : 221f '() g '() e e 1= = = (1) 1g() f() f '()( ) e e ( ) = = 1e e e = ( ) 21 1 1 1 2 2e e e e e e e 1 = = = = ( ) ( ) ( ) ( )(1)2 2 2 22 2 2 2 24e 1 e 1 4 e e 1 4 e 1 4 = = = = (i) , , (,f()) , (,f()) Cf Cg . 26) 1f, f 1fC yy 1 1y 0f (y) 1lim 13y= . fC (1,f(1)). 1fC yy 1 1f (0) 1 0 f(1) = = . 1f (y) x y f(x) = = 1 1y 0 y 0lim x lim f (y) f (0) 1 = = = ( 1f 0 0) 1 1 1y 0 y 0 y 0y 0f (y) 1 f (y) f (0) x 1lim 13 lim 13 lim 13y y f(x) 01 1 1lim 13 13 f '(1)f(x) f(1) f '(1) 13x 1 = = = = = = , fC (1,f(1)) (): 1y f(1) f '(1)(x 1) y (x 1)13 = =

mathhmagic.blogspot.gr : ., ., ., ., . 53 27) f : 23 x xf (x) 3f(x) e x 12+ = + x i) xg(x) e x 1= + , g(x) 0= g. ii) f. iii) f. iv) f . i) ( )x xg '(x) e x 1 ' e 1 0= + = + > , g ., g(0) 0= , x 0= g(x) 0= . x 0= g(x) 0= , g . x 0< g(x) g(0) g(x) 0< < x 0> g(x) g(0) g(x) 0> > , g ( ),0 ( )0,+ . ii) f . f , f '() 0= . 23 x xf (x) 3f(x) e x 12+ = + : ( ) ( )2 x 2 x 23f (x)f '(x) 3f '(x) e x 1 f '(x) 3f (x) 3 e x 1 f '(x) 3f (x) 3 g(x)+ = + + = + + = (1) (1) x = ( )2f '() 3f () 3 g() g() 0+ = = f '() 0= . g() 0 0= = , (i). , f x 0= . iii) f ( f ) . f(0). x 0< (1) ( )2f '(x) 3f (x) 3 g(x) 0+ = < f '(x) 0< ( ),0 x 0> (1) ( )2f '(x) 3f (x) 3 g(x) 0+ = > f '(x) 0> ( )0,+ , f(0) f . x 0= ( )23 0 20f (0) 3f(0) e 0 1 f(0) f (0) 3 0 f(0) 02+ = + + = = f(0) 0= f. iv) f ( ,0 )0, + , f(0) 0= f.

mathhmagic.blogspot.gr : ., ., ., ., . 5428) : , xf(x) e 2016x,x 1,2= + 0x 2= . : f 0x 2= . f 1,2 , xf '(x) e 2016= + . Fermat 22 ef '(2) 0 e 2016 0 02016= + = = < f 2. : , =5>0 xf(x) 5e 2016x,= + xf(x) 5e 2016 0= + > x 1,2 f 1,2 . 0x 2= . , 0> f 0x 2= ; . f . Fermat . 29)( all time classic..) ( )2f(x) x x ln x,x 0,= + + . ii) 2g(x) 1 x x= + h(x) 1 ln x= + , . iii) 2 3x x2 xln x x2 3+ > + ( )x 0, + i) 2l 2x x lf '(x) 2x 1x x += + = ( )x 0, + 22x x l 0 + > ( )x 0, + ( =-7

mathhmagic.blogspot.gr : ., ., ., ., . 55ii) 2h(x) g(x) h(x) g(x) 0 x x ln x 0 f(x) 0= = + = = . (i) f(x) 0= 1. g h (1,1).: ( )2g '(x) 1 x x ' 1 2x= + = 1h '(x)x= g '(1) 1= h'(1) 1= g '(1)h '(1) 1= Cg,Ch (1,1) -1 . iii) 2 3x x(x) 2 x ln x x ,x 02 3 = + + > 2 32 2x x'(x) 2 x ln x x ' ln x 1 1 x x ln x x x f(x)2 3 = + + = + + = + = x 0> (x) f(x) 0x 1= 5(1)6 = . , ( )x 0, + : 5(x) (1) (x) 06 > 2 3x x2 x ln x x 02 3+ + > 2 3x x2 x ln x x2 3+ > + ( ) '( )x f x = - + x 0 1 + ( )x

mathhmagic.blogspot.gr : ., ., ., ., . 5630) f : : 23 2x 1ln x (x 1) ln xf( 1) 2 limln x (x 1)+ + =+ 3xf '(x) e= x f . i) x 1ln xlimx 1 ii) (i) f( 1) 0 = . iii) e= . iv) f . v) f . i) g(x) ln xx 1 x 1 x 1g(x) g(1)ln x ln x ln1lim lim lim g'(1)x 1 x 1 x 1= = = = 1g '(x)x= g '(1) 1= x 1ln xlim 1x 1= ii)2 22 2 2 23 2 3 2 3 2x 1 x 1 x 12 2 2ln x (x 1) ln x ln x (x 1)ln xln x (x 1)ln x (x 1) (x 1) (x 1)lim lim limln x (x 1) ln x (x 1) ln x (x 1)(x 1) (x 1) (x 1) + ++ = = =+ + + 2222 (i)3 2 2x 1 x 12 2ln x ln x ln x ln xx 1 x 1 x 1 1 1(x 1)lim lim 2ln x ln x 0 1 11 ln x 1(x 1) (x 1) + + + = = = = ++ + f( 1) 2 2 f( 1) 0 + = = iii) f f(x) 0 x f(x) f( 1) x f 1x 1= . f 1x 1= Fermat : f '( 1) 0 = : 3( 1)f '( 1) 0 e 0 e = = = iv) e= : 3xf '(x) e e= x x3 3ex 1 x 3f '(x) 0 e e 0 e e 1 x x 1 > > > > > x3 3ex 1 x 3f '(x) 0 e e 0 e e 1 x x 1 < < < < < f '(x) 0 ... x 1< = f ( , 1 f )1, + f f( 1) 0 = v) 3xf '(x) e e= , f 32 xf ''(x) .. 3x e 0= = > x

mathhmagic.blogspot.gr : ., ., ., ., . 57 f . 31) . , , , , . 1) f : xf(x)lim x+= ( )xlim f(x) x + = y x = + Cf + . 2) f : f ''(x) 0< x . 3) f , f. 4) f,g : , f(x) g(x) x , f '(x) g '(x)< x . . 0x xf '(x)limg'(x) , , 0x xf(x)limg(x)( 00). . f , : i) ii) iii) .1) 2) 3) 4) . , LHospital . . (ii) .. f(x T) f(x)+ = x A , T 0 : ( ) ( )f(x T) ' f(x) ' f '(x T)(x T)' f '(x) f '(x T) f '(x)+ = + + = + = x A , T 0 f . 32) f : (1) 2xf(x) f '(0) f '(x) x ( )2f ''(0) f(0) 1+ x 0> (1) f '(0) f '(x) f '(0) f '(x) f '(x) f '(0)2xf(x) f '(0) f '(x) 2f(x) 2f(x) 2f(x)x x 0 x 0 ( )x 0 x 0f '(x) f '(0)lim 2f(x) lim 2f(0) f ''(0)x 0 ( f )

mathhmagic.blogspot.gr : ., ., ., ., . 58 x 0< (1) f '(x) f '(0)2xf(x) f '(0) f '(x) ... 2f(x)x 0 ( )x 0 x 0f '(x) f '(0)lim 2f(x) lim 2f(0) f ''(0)x 0 2f(0) f ''(0)= ( ) ( ) ( )+ + = + + = 2 2 2f ''(0) f(0) 1 2f(0) f(0) 1 f(0) 1 0 ( )2f ''(0) f(0) 1+ ( Fermat) 33) : x 1f(x) lnxx 1+= , x 0> x 1 g(x) lnx= A(,ln ) , 0> , xh(x) e= B(,e ) , , , f(x) 0= .( 2006) Cg A(,ln ) : 11 1( ) : y ln (x ) y x 1 ln = = + Cf B(,e ) : 2( ) : y e e (x ) y e x e e y e x e (1 ) = = + = + (1 2( ),( ) 1 ln ln ln e 1 1 1 1(1 ) 1 ln (1 ln) 1 ln ln 1 ln e (1 ) 1 ln = = == = + + = + + = + = + ( )ln ln ln 1 1 ln 1 ln ln 1 ln ln = = = + = ++ = + + = + ln ln ln 1 1 f() 0ln ln 01 1 = = = + +== = + f(x) 0= . 34) f : f(0) 1= : 2 2h 0f(x 2h) f(x h) 2xf(x)limh 3h x 1+ =+ + x i) :

mathhmagic.blogspot.gr : ., ., ., ., . 59 2f(x 2h) f(x h) 2 f(x 2h) f(x) 1 f(x h) f(x)h 3 2h h 3 hh 3h+ + = ++ + + x , h 0 h 3 ii) 22xf(x)f '(x)x 1=+, x . i) x , h 0 h 3 2f(x 2h) f(x h) f(x 2h) f(x) f(x) f(x h) f(x 2h) f(x) f(x h) f(x)h(h 3) h(h 3) h(h 3)h 3hf(x 2h) f(x) f(x h) f(x) 2 f(x 2h) f(x) 1 f(x h) f(x)h(h 3) h(h 3) h 3 2h h 3 h2 f(x 2h) f(x) 1 f(x h) f(x)h 3 2h h 3 h+ + + + += = + =+ + +++ + = = =+ + + ++ = ++ + ii) 2h uh 0 h 0 h 0 h 0 u 02 f(x 2h) f(x) 2 f(x 2h) f(x) 2 f(x u) f(x) 2lim lim lim lim f '(x)h 3 2h h 3 2h 3 u 3= + + + = = = + + h uh 0 h 0 h 0 u 0 u 01 f(x h) f(x) 1 f(x h) f(x) 1 f(x u) f(x) 1lim lim lim lim f '(x)h 3 h h 3 h 3 u 3 = + = = = + + 2h 0f(x 2h) f(x h) 2 1lim f '(x) f '(x) f '(x)3 3h 3h+ = + =+ , x 22xf(x)f '(x)x 1=+, x . 35) f : , ) , . ) ( ), . ) 2 2f() f() = . ( )0x , 0 0f '(x ) 2x= . 0 0f '(x ) 2x 0 = 2f(x) x . g : , 2g(x) f(x) x= (1) g : i) , f 2x . ii) ( ), ( ), f 2x . iii) 2 2g() f() f() g()= = = . Rolle ( )0x , : 0g '(x ) 0= . (1)g '(x) f '(x) 2x= (1)0 0 0g '(x ) f '(x ) 2x 0= = 0 0f '(x ) 2x=

mathhmagic.blogspot.gr : ., ., ., ., . 6036) Rolle - . - Rolle !! : -; - f 3f(x) 3x= 1,1 . f 1,1 1,1 . : 2f '(x) 9x= . ( ) 1,1 : 2f '() 9 0= = 0= Rolle . 3 3f( 1) 3( 1) 3(1) f(1) = = Rolle !! ; Rolle . f(x) . ( Rolle,.. G.Aligniac) 37)) , f,g ( ), f(x) 0> x , ln f() ln f() g() g() = . ( ) , , f '() f()g '() 0+ = . ) f : , ( 0> ) ( ), , f() f( )f(0)2+ = (1). ( ) , , f ''() 0= . ) f : . f f . f 1-1. ) f 0,1 0,1 , f(0) 0= f(x) 0> ( )x 0,1 . ( ) 0,1 f '() f '(1 )2f() f(1 )= ) ln f() ln f() g() g() ln f() g() ln f() g() = + = + (1) h(x) ln f(x) g(x)= + Rolle. h , ( ), , h() h()= ( ) , , h'() 0= .

mathhmagic.blogspot.gr : ., ., ., ., . 61 = +f '(x)h '(x) g '(x)f(x) = + = + =f '()h '() 0 g '() 0 f '() f()g '() 0f() ) f ,0 , 0, . ( )1 ,0 , ( )2 0, , : (1)1f() f( )f( )f(0) f( ) f() f( )2f '( ) (2)0 ( ) 2+ = = = (1)2f() f( )f()f() f(0) f() f( )2f '( ) (3) 0 2+ = = = f 1 2 , . Rolle( f 1 2 , , 1 2f '( ) f '( )= ) ( ) ( )1 2 , , , f ''() 0= . ) f 1-1. 1 2x ,x , 1 2x x< 1 2f '(x ) f '(x )= . f , f '' . Rolle f 1 2x ,x ( f 1 2x ,x 1 2f '(x ) f '(x )= ). ( )1 2 x ,x f ''() 0= . f , , f.. f 1-1. ) ( )2g(x) f(x) f(1 x)= 0,1 . g 0,1 , ( ). .. ( ) 0,1 , g(1) g(0)g '() (1)1 0= ( )( ) ( ) ( )2 2 2g '(x) f(x) f(1 x) ' 2f(x)f '(x)f(1 x) f(x) f '(1 x)(1 x)' 2f(x)f '(x)f(1 x) f(x) f '(1 x)= = + = ( )2g '(x) 2f(x)f '(x)f(1 x) f(x) f '(1 x)= ( )2g '() 2f()f '()f(1 ) f() f '(1 )= : ( ) ( )2 2g(1) f(1) f(1 1) f(1) f(0)= = , ( ) ( )2 2g(0) f(0) f(1 0) f(0) f(1)= = (1) : ( ) ( ) ( ) ( )2 2f(0) 02 2f(1) f(0) f(0) f(1)g(1) g(0)g '() f(1) f(0) f(0) f(1) 01 0 1== = = =( ) ( )( )2 22f() f '(1 )g '() 0 2f()f '()f(1 ) f() f '(1 ) 0 2f()f '()f(1 ) f() f '(1 ) 2 f '()f(1 )f()= = = =( )2f() f '(1 ) f '() f '(1 )2 f '() 2f(1 ) f() f(1 )f() = =

mathhmagic.blogspot.gr : ., ., ., ., . 6238) f : , , f() f() 0= = (1) i) g, 0f(x)g(x)x x= 0x , ( ) , , g '() 0= ii) () f (,f()) 0N(x ,0) . i) g , , .: ( )020f '(x) x x f(x)g '(x)(x x ) = (2) g ( ), . : (1)0f()g() 0x x= = , (1)0f()g() 0x x= = Rolle , ( ) , , g '() 0= (3) ii) (3) : ( ) ( ) ( )00200f '() x f() f()0 f '() x f() 0 f '() (4) x( x ) = = = () (,f()) f : ( ) ( ) ( )(4)0f()y f() f '() x y f() x (5) x = = 0(x ,0) (5).: ( ) ( )00f()0 f() x f() f() x = = () 0N(x ,0) . 39) f,g g . , x ,: g '(x) g(x) 0+ = f(x) g(x) 1 = i) 1( ) f (,f()) xx. ii) 2( ) f B(,g()) xx. iii) 1( ) 2( ) , .

mathhmagic.blogspot.gr : ., ., ., ., . 63 i) x ,: ( )x x x xg '(x) g(x) 0 e g '(x) e g(x) 0 e g(x) ' 0 e g(x) c+ = + = = = x xxce g(x) c g(x) g(x) cee = = = , c 0, c 0= , g(x) 0= , f(x)g(x) 0= , . xx1 1 1f(x)g(x) 1 f(x) f(x) f(x) eg(x) cce= = = = x , x1f '(x) ec= . (,f()) : ( ) 11 1 : y f() f '()(x ) y e e (x )c c = = y=0, 1 10 e e (x ) 1 x x 1c c = = = . ( )1 xx ( 1,0) ii) x , xg '(x) ce= . B(,g()) : ( ) 2 : y g() g '()(x ) y ce ce (x ) = = y=0, ( )2 : 0 g() g '()(x ) 1 x x 1 = = = + . ( )2 xx ( 1,0) + iii) ( ) 1g '()f '() ce e 1c = = , . 2 1 5 3 4 0 y x Cf !!

mathhmagic.blogspot.gr : ., ., ., ., . 6440) , . : ) f(x) g(x)x = Rolle , ) Cf. ) g , ( ), . f() g() = = ( , ) f() g() = =( , ) g() g()= Rolle , ) Rolle , ( )0x , 0g '(x ) 0= (1) 2(f(x) )'(x ) (f(x) )(x )'f(x) g '(x) 'x (x ) = = = 2f '(x)(x ) (f(x) )(x ) = (1) : 0 0 00 0 0200 0 0 0 0 0f '(x )(x ) (f(x ) )0 f '(x )(x ) (f(x ) ) 0(x )(f(x ) ) f '(x )(x ) f(x ) f '(x )( x ) = = = = 0 0 0y f(x ) f '(x )(x x ) = ( ) . (,) () Cf y x O

mathhmagic.blogspot.gr : ., ., ., ., . 6541) f 1,4 , ( )1,4 f ( )1,4 . f(2) f(3)+ f(1) f(4)+ . .. 1,2 3,4 . f 1,4 1,2 3,4 . f ( )1,4 ( )1,2 ( )3,4 . .. f 1,2 3,4 : ( )1x 1,2 : 1f(2) f(1)f '(x ) f(2) f(1)2 1= = (1) ( )2x 3,4 : 2f(4) f(3)f '(x ) f(4) f(3)4 3= = (2) f ( )1,4 1 2x x< 1 2f '(x ) f '(x ) f(2) f(1) f(4) f(3) f(2) f(3) f(1) f(4)> > + > + 42) f , , ( ), 0> f '(x) < ( )x , ( )1 2 , , 1 2 : f f1 2 1 2( ) ( ) ( )1 2 , , 1 2 . 1 2 < 1 2 > . 1 2 < .. 1 2 , .: f 1 2 , , f ( ) ( )1 2 , , ( ) , : 2 1 2 1f( ) f( ) f '()( ) = 1 2 1 2f( ) f( ) f '() < f '() < 1 2 1 2f( ) f( ) < . 43) f i) f (4,17) -6 4%, . ii) x>12; i)f(4)=-6 ( ) ii) f . B(10,0) A(4,17) % % x y 4 12

mathhmagic.blogspot.gr : ., ., ., ., . 66 44) f : , (,f()) B(,f()) Cf. ) Cf (,f()) , ( ) , , ( )0x , 0f ''(x ) 0= . ) Cf (,f()) , ( ) , , ( )1 2 , , , 1 2f '( ) f '( ) 1 = . ) ,, , f() f()f() f() (1) = = .. f , , ( )1x , ( )2x , : 1 2f() f()f() f()f '(x ) , f '(x ) = = (1) : 1 2f '(x ) f '(x )= . Rolle f 1 2x ,x . ) f() f()f() f() 1 1 = = .. , , , . 45) i) f(x) x= x, x + + , , > , x max{ , }> ii) (i) , xlim ( x ) ( x )+ + + i) x max{ , }> f(x) x= x, x + + ( ) x, x+ + .. ( ) x, x + + f( x) f( x)f '() x x+ +=+ + ii) ( )( x) ( x) ( x) ( x)f '() x x x x x x ( x) ( x)+ + + += = + + + ++ + = + +

mathhmagic.blogspot.gr : ., ., ., ., . 67( ) ( ) ( x) ( x) x x x x x x x x> + + = + + + + = + + =+ + + x lim 0 x x+=+ + + xlim ( x ) ( x ) 0+ + + = 46) ! . - ! -; f f-1 f '(x) 0> x . 1f fC ,C (,), .: f()= 1 f ()= f()= f()= , , ( ) , , : f() f()f '() = f '() 1 0 = = < f '() 0< x f '(x) 0> x . ; 1 -3 , = . ( ), 47)( ) f : , ( ), : i) f '(x) 0> ( )x , , f() f()> ii) f '(x) 0< ( )x , , f() f()< i) , ( ) , f() f()f '() =. f '() 0> ->0 f() f() 0 > f() f()> . (ii)

mathhmagic.blogspot.gr : ., ., ., ., . 6848) (ll time classic) 2 2f(x) 2x ln x 3x 4x 1= + i) f f. ii) f . iii) fC . iv) f . v) f. vi) Cf . vii) f. i) f ( )A 0,= + : f '(x) .. 4x ln x 4x 4= = + x 0> . f ''(x) .. 4 ln x= = x 0> . ii) f ( )0,1 , ( )1,+ f 0x 1= , f : ( )0,1 )1, + iii) f ''(1) 0= f 1, Cf A(1,f(1)) A(1,0). iv) f (0,1 )1, + .: x 1 f '(x) f '(1) f '(x) 0< > > x 1 f '(x) f '(1) f '(x) 0> > > f '(x) 0> x 1 . f ( )A 0,= + f . v) f . : x 1 f(x) f(1) f(x) 0< < < x 1 f(x) f(1) f(x) 0> > > , f ( )0,1 ( )1,+ . vi) f f(1)=0, f(x) 0= x 1= ., (1,0) vii): - + ''( )f x x 0 1 + ( )f x . + + '( )f x x 0 1 + ( )f x

mathhmagic.blogspot.gr : ., ., ., ., . 69xlim f(x) ..+= = + ( )2 2x 0 x 0lim f(x) lim 2x ln x 3x 4x 1 1 = + = , ( )0( )2 2x 0 x 0 x 0 D.H.L x 0 x 02 422 ln x 1xlim f(x) lim 2x ln x lim lim lim x 01 2x 2x x = = = = = f f(A) ( 1, )= + . 49)( ..) f : x : f(x) 4 3 2f(x) e x 4x 12x 8+ = + + (1) : ) Cf . ) f . (1) : f(x) 3 2f '(x) e f '(x) 4x 12x 24x+ = + (2) ( )( )2f(x) f(x) 22f(x) f(x) 2f ''(x) e f '(x) e f ''(x) 12x 24x 24f ''(x) e f '(x) e f ''(x) 12(x 2x 2) (3)+ + = + + + = + ) Cf 0 0A(x ,f(x )) , 0f ''(x ) 0= (3) 0x x= ( ) ( )0 0 02 2f(x ) f(x ) f(x )2 20 0 0 0 0 0 0 0f ''(x ) e f '(x ) e f ''(x ) 12(x 2x 2) e f '(x ) 12(x 2x 2)+ + = + = + ( )0 2f(x ) 0e f '(x ) 0 20 012(x 2x 2) 0 + < ( 2y 2y 2 + =1>0, 2y 2y 2 0 + > y ). ) (2) : f ( x ) 3 21 e 0f(x) 3 2 f(x) 3 2f(x)4x 12x 24xf '(x) e f '(x) 4x 12x 24x f '(x)(1 e ) 4x 12x 24x f '(x1 e+ + + = + + = + =+ 3 2 2 2f '(x) 0 4x 12x 24x 0 4x(x 3x 6) 0 x 0= + = + = = . f x 0= , x 0= f.

mathhmagic.blogspot.gr : ., ., ., ., . 7050)( ) , ( )x , f() f(x)f(x) f()x x< ,x , x, f .. , ( ) ( )1 2 ,x , x, 1 2f() f(x)f() f(x)f '( ) , f '( ) x x= = f f ' f '1 2 1 2 f( ) f( )< < f() f(x)f(x) f()x x< . 51)(1 1997) ) f . : ( ) ( )1 21 2 f x f xx xf2 2+ + , 1 2x ,x ) g, : g(x) 0> 2g ''(x) g(x) g '(x) 0 > , x . : i) g'g . ii) ( ) ( )1 2 1 2x xg g x g x2 + 1 2x ,x . ) 1 2x x , 1 2x x< f : 1 2 1 21 2x x x xx , , ,x2 2 + + : 1 2 1 21 1 2 2x x x x x , , ,x2 2 + + ( )( ) ( )1 2 1 21 111 2 2 11x x x xf f x f f x2 2f ' x x x xx2 2 + + = =+ ( )( ) ( )1 2 1 22 221 2 2 12x x x xf x f f x f2 2f ' x x x xx2 2 + + = =+ f : ( ) ( )( ) ( )1 2 1 21 2f '1 2 1 21 2 1 2 1 22 1 2 1x x x xf f x f x f2 2 x x x x f '( ) f '( ) f f x f x fx x x x 2 22 2 + + + + < < < < x , x2+= , x x 0 = > : f() f()f2 2 + +

mathhmagic.blogspot.gr : ., ., ., ., . 71( ) ( ) ( ) ( )2 11 2 1 22 1f x f xx x x x2f f x f x f2 2 2+ + + < + . 4) f , . 1) 2) 3) 4) 5)

mathhmagic.blogspot.gr : ., ., ., ., . 7253) f : , 0x 0 : f ''(x) 4(f '(x) f(x))> x ) : 2xg(x) f(x) e= . ) f(x) 0 x .( ) ) x : ( )2x 2x 2x 2x 2xg'(x) f(x) e ' f '(x) e f(x) e ( 2x)' f '(x) e f(x) 2e = = + = ( ) ( ) ( )2x 2x 2x 2x 2x 2xg''(x) f '(x) e f(x) 2e ' f ''(x) e f '(x) e ' f '(x) 2e f(x) 2e ( 2x)' = = + + = 2x 2x 2x 2x 2xf ''(x) e 2f '(x)e f '(x) 2e f(x) 4e e (f ''(x) 4f '(x) 4f(x)) 0 + = + > )H f 0x , 0x 0x . Fermat 0f '(x ) 0= , 0f(x ) 0= . g(x) 0 x g , g , : 0 0x x g '(x) g '(x ) g '(x) 0< < < 0 0x x g '(x) g '(x ) g '(x) 0> > > g ( )0,x )0x , + , 0x . x : 0g(x) g(x ) g(x) 0 f(x) 0 x . 54) ( )f : 0,+ , 2 2ln x ln x 2x 2f(x)x+ + += , f(x) 2 2 + ( )x 0, + . i) f(1) 2= . ii) f . iii) f . iv) ( )g : 0,+ , Cf + . g(x) xg '(x)= ( )0,+ . i) 2 2ln 1 ln1 2 1 2f(1) 2 21+ + += = + ( ) ( )( )2 2 2 22 22ln x ln x 2x 2 'x ln x ln x 2x 2 x 'ln x ln x 2x 2f '(x) 'x x+ + + + + + + + += = = ( )2 2 2 2 22 21 12 ln x 4x x ln x ln x 2x 2x x 2 ln x 4x ln x ln x 2x 2x x + + + + + + + = = 2 222 ln x 2x ln x x+

mathhmagic.blogspot.gr : ., ., ., ., . 73 ( )x 0, + : f(x) 2 2 f(x) f(1) + f 0x 1= , 1 ( )0,+ f 1. Fermat 2 222 ln1 2 1 ln 1 f '(1) 0 0 2 0 21+ = = = = ii) 2 222 ln x 2x ln x 2f '(x)x+ = ( )232 ln x ln x 2f ''(x) 0x += > x 0> iii)H f f '(1) 0= . f '(x) f iv) : 2 2 2 2(*)2 2 2 2x x x x xf(x) ln x 2 ln x 2x 4 ln x 2 ln x 2x 4lim lim lim lim limx x x x x0 0 2 2+ + + + ++ + + += = + + == + + = (*)x D.H.L x1ln x xlim lim 0x 1+ += = 2xln xlim 0x+ = ,2x2 ln xlim .. 0x+= = ( )xlim f(x) 2x .. 0+ = = Cf + : : y 2x= ( )1 2x ,x 0, + , 1 2x x< 1 1g(x ) 2x= 2 2g(x ) 2x= g(x)h(x)x= , 1 2x ,x , ( )1 2x ,x 1 11 21 1g(x ) 2xh(x ) 2 , h(x ) ... 2x x= = = = = Rolle ( )3x 0, + 3h'(x ) 0= , 2g(x) g '(x) x g(x)h '(x) 'x x = = 3 3 33 3 3 3 3 323g '(x ) x g(x )0 g '(x ) x g(x ) 0 g '(x ) x g(x )x = = = . - + '( )f x x 0 1 + ( )f x

mathhmagic.blogspot.gr : ., ., ., ., . 7455) x x x ex e 0 ( )x 0, + , >0 i) . ii) ( )xx 0lim 1 x ++ = . i) x x x ef(x) x e = e e e e e e e e 0f(e) e e e e 1 e 1 e 1 1 1 0 = = = = = = , f(x) f(e) ( )x 0, + . f 0x e= 0x e= ( )0,+ Fermat f '(e) 0= . (1) ( ) ( ) ( ) ( )xx x x e x x x x x e ln x x x x x ef '(x) x e ' x 'e x e ' ln (x e)' e 'e x e ( x)' ln = = + = + = ( ) ( )( ) ( )( )x ln x x x x x e x ln x x x x x e x ln x x x x x ee 'e x e ln e xln x ' e x e ln e ln x 1 e x e ln = = + = ( )( )x x x x x ex ln x 1 e x e ln = + (1) ( )( ) ( )( )e e e e e e e e e ee ln e 1 e e e ln 0 e 1 1 e 1 ln 0 2e e 1 ln 0 + = + = =2 1 ln 0 ln 1 e = = = ii)To ( )xx 0lim 1 x++ 1 . ( ) ( ) ( )xx ln 1 x x ln 1 xx 0 x 0 x 0lim 1 x lim e lim e+ + ++ + + = = (2) ( )x 0lim x ln 1 x++ ( ) 0+ ( ) ( )020D.L.Hospitalx 0 x 0 x 0 x 0 x 0 x 021ln(1 x) ln(1 x) (ln(1 x))' x1 xlim xln 1 x lim lim lim lim lim 11 1x 1 xx 'x x+ + + + + + + + + + + = = = = = = + , (2) ( ) ( ) ( )xx ln 1 x x ln 1 x 1x 0 x 0 x 0lim 1 x lim e lim e e e+ + ++ + + = = = = 56) f : 2xf(x) f '(0) f '(x) , x 2f ''(0) f (0) 1+ , x 2xf(x) f '(0) f '(x) 2xf(x) f '(x) f '(0) 0 + (1) h(x) 2xf(x) f '(x) f '(0)= + , x h(0) 2 0 f(0) f '(0) f '(0) 0= + = , (1) : h(x) h(0) x , h 0 . Fermat h'(0) 0= h. h'(x) 2f(x) 2xf '(x) f ''(x)= + + , x Fermat ..

mathhmagic.blogspot.gr : ., ., ., ., . 75 h'(0) 0 2f(0) 2 0f '(0) f ''(0) 0 2f(0) f ''(0) 0 f ''(0) 2f(0)= + + = + = = , : ( )f ''(0) 2f(0)22 2 2f ''(0) f (0) 1 2f(0) f (0) 1 1 2f(0) f (0) 0 1 f(0) 0=+ + + x , . 57) f : , : x 1f(x) 1lim 1x 1= (1) f '(x) 1 x (2) f ) f(1),f(1) Cf 1. ) f. ) x 11974limx f(x) ) 1 , f(x) 1g(x)x 1= ( ) ( )f(x) 1g(x) g(x) x 1 f(x) 1 g(x) x 1 1 f(x)x 1= = + = ( )( )x 1 x 1 x 1lim g(x) x 1 1 lim f(x) lim f(x) 1 + = = f 1 f 1 x 1f(1) lim f(x) 1= = x 1 x 1f(x) 1 f(x) f(1)lim 1 lim 1 f '(1) 1x 1 x 1 = = = (1,f(1) ) : : y f(1) f '(1)(x 1) = : y 1 1(x 1) = : y x= ) f '(x) 1 x f '(x) f '(1) x f 0x 1= Fermat f ''(1) 0= . f 0x 1= f ''(x) 0= . f x 1 f ''(x) f ''(1) f ''(x) 0< < < f ( ),1 f 'x 1 f '(x) f '(1)< > (2) f . f fff ''(x) 0 x 1f ''(x) 0 f ''(x) f ''(1) x 1f ''(x) 0 f ''(x) f ''(1) x 1= =< < >> > < ) ) ff ''(x) 0 f ''(x) f ''(1) x 1< < > f x 1> Cf (1,f(1)) f(x) x< x 1> += +x 11974limx f(x)

mathhmagic.blogspot.gr : ., ., ., ., . 76 = x 11974limx f(x) . ( ) f , , , Cf. f , Cf. 0x x> . 0 0M(x ,f(x )) 0 0 0y f(x ) f '(x )(x x ) = 0 0 0y f '(x )(x x ) f(x )= + Cf A(x,y) B(x,f(x)) . : 0 0 000 0 0 00y f(x) f '(x )(x x ) f(x ) f(x)f(x) f(x )f '(x )(x x ) f(x) f(x ) f '(x ) (1)x x> + > > > 0x ,x f , , , ( )0 x ,x , 00f(x) f(x )f '()x x= (2) (2) , (1) , : 0f '(x ) f '()> , f f : f '0 0x f '(x ) f '()< > 58) f : 0,1 , , f(x) 0,f(0) 1,f '(0) 0> = = : ( )2 3f(x)f ''(x) 2 f '(x) f(x) = , x 0,1 ) g 2f '(x)g(x) xf (x)= 0,1 . ) f. ) f 0,1 . ) f 0,1 . ) : 2 2 2 (1)4 3f ''(x)f (x) 2(f '(x)) f(x) f ''(x)f(x) 2(f '(x))g '(x) 1 1 1 1 0f (x) f (x) = = = = g(x) c= (2) x 0,1 A(x, y) 0 0M(x ,f(x )) x0 x B(x,f(x))

mathhmagic.blogspot.gr : ., ., ., ., . 77) x 0= (2) 2f '(0)g(0) c 0 c c 0f (0)= = = . g(x) 0= x 0,1 .: 2 22 2f '(x) f '(x) 1 x 1 xg(x) 0 x 0 x ' ' c,f(x) 2 f(x) 2f (x) f (x) = = = = = + x 0,1 x 0= 1 c = , : 221 x 21 .... f(x)f(x) 2 2 x = =, x 0,1 ) ( ) = = = > 2 222 4xf '(x) ' .. 02 x 2 x x 0,1 . f 0,1 . ) : ( ) ( ) + = = = > 22 32 24x 4(3x 2)f ''(x) ' .. 02 x 2 x x 0,1 . f 0,1 . 59) f : : f(0) 1= 6xf(x)f '(x) 3e= x f. ( ) ( )6x 6x 2 6x 2 6xf(x)f '(x) 3e 2f(x)f '(x) 6e f (x) ' e ' f (x) e c= = = = + (1) x=0 (1) : 2 6 0f (0) e c 1 1 c c 0= + = + = 2 6xf (x) e= (2) x (2) f . , . f(0) 1 0= > , f(x) 0> x . 2 6x 3xf (x) e f(x) e= = x Bonus. - : (2) : ( )( )( )22 6x 2 6x 2 3x3x 3x 3x 3x 3x 3xf (x) e f (x) e 0 f (x) e 0f(x) e f(x) e 0 f(x) e 0 f(x) e 0 f(x) e f(x) e= = = + = = + = = = x=0 3 0f(0) e 1= = 3xf(x) e= . ; f,g f g 0 = f,g f 0= g 0= .

mathhmagic.blogspot.gr : ., ., ., ., . 7860) ( ) ( )f : 0, 0,+ + , f ( )0,+ f '(x) f(x) ln(xe)= (1) ( )x 0, + f ( )0x 0, + : e0 0f(x ) x = (2) x 0> (1) ( ) ( ) ( )f(x) 0 f '(x)f '(x) f(x)ln(xe) ln x ln e ln f(x) ' ln x 1 ln f(x) ' x ln x 'f(x)>= = + = + = ln f(x) x ln x c, x 0= + > (3). c. ( )0x 0, + .Fermat 0f '(x ) 0= (1) ( )x 0, + 0x x= 0f(x ) 00 0 0 0 0 0 0 01f '(x ) f(x )ln(x e) 0 f(x )ln(x e) ln(x e) 0 x e 1 xe>= = = = = (2) e e1 1 1 1 1 1 1f( ) ln f( ) ln e ln f( ) 1 ln f( )e e e e e e e = = = = (4) (3) 1xe= (4)1 1 1 1 1ln f( ) ln c c c 0e e e e e= + = + = xx ln x ln x xln f(x) x ln x f(x) e f(x) e f(x) x , x 0= = = = > . 61)( 1974 ) 197319741974 11974 1+ =+ 197419751974 11974 1+ =+ ) xx 11974 1f(x)1974 1++=+, x + += >+ + f 1973 19741974 19751974 1 1974 11973 1974 f(1973) f(1974)1974 1 1974 1 62)( .. ) f : f(4) 3= : x 1f(x) 3xlim 2x 1= i) f(1) 3= ii) f '(1) Cf A(1,f(1)) . iii) y x 1= + Cf ( )0x 1,4 . iv) f , : ) ( ) 1,4 f ) f(3) f(6)< .

mathhmagic.blogspot.gr : ., ., ., ., . 79i) x 1f(x) 3xlim 2x 1=. ( ) ( )f(x) 3xg(x) g(x) x 1 f(x) 3x f(x) g(x) x 1 3xx 1= = = + f 0x 1= , : ( )( ) ( ) ( )x 1 x 1 x 1 x 1 x 1f(1) lim f(x) lim g(x) x 1 3x limg(x)lim x 1 lim 3x 3 = = + = + = ii) ( ) ( ) ( )x 1 x 1 x 1 x 1g(x) x 1 3x f(1) g(x) x 1 3x 3 g(x) x 1 3(x 1)f(x) f(1)f '(1) lim lim lim limx 1 x 1 x 1 x 1 + + + = = = = = ( ) ( )( ) ( )x 1 x 1 x 1g(x) x 1 3(x 1) x 1 g(x) 3lim lim lim g(x) 3 5x 1 x 1 + += = = + = Cf 0x 1= : y f(1) f '(1) (x 1) ... y 5x 2 = = = iii) f(x) x 1= + ( )1,4 . h(x) f(x) x 1= 1,4 : h(1) f(1) 1 1 3 1 1 1= = = h(4) f(4) 4 1 3 4 1 2= = = h(1)h(4) 0< Bolzano (1,4) . f y=x+1. iv)) f f f ''(x) 0> f(1) f(4)= , f 1,4 , f ( )1,4 Rolle ( ) 1,4 f '() 0= . f . f. f 'x f '(x) f '() 0< < = f 'x f '(x) f '() 0> > = , f . f ( ) ( ), , , , f . ) f 1,3 , 4,6 ( ) ( )1 2x 1,3 ,x 4,6 : 1f(3) f(1) f(3) 3f '(x )3 1 2 = = 2f(6) f(4) f(6) 3f '(x )6 4 2 = = f f .

mathhmagic.blogspot.gr : ., ., ., ., . 80: 1 2f(3) 3 f(6) 3f '(x ) f '(x ) ... f(3) f(6)2 2 < < < 64) f (1) ( )3 3f(x) f(x) x 1+ = + x ) f 1-1. ) ( )32 3f(2x 1) f(x) x 1 + = + . f , : ) f f . ) f A(1,f(1)) . ) 1,1 = f(A) . ) 3 f(x)2= . ) ( )3 3f(x) f(x) x 1+ = + , x 1 2x ,x 1 2f(x ) f(x )= , 3 31 2f(x ) f(x )= ( ) ( )(1)3 3 3 31 1 2 2 1 2 1 2f(x ) f(x ) f(x ) f(x ) x 1 x 1 x x+ = + + = + = f 1-1. ) ( ) ( ) ( ) ( ) ( )(1)3 3 33 32 3 2 2f(2x 1) f(x) x 1 f(2x 1) f(x) f(x) f(x) f(2x 1) f(x) + = + + = + = ( ) ( )f 1 13 32 2 2 2 1f(2x 1) f(x) f(2x 1) f(x) 2x 1 x 2x x 1 0 x 1 x2 = = = = = = ) x : ( )( ) ( ) ( ) ( )3 2 23 2 2f(x) f(x) ' x 1 ' 3 f(x) f '(x) f '(x) 3x f '(x) 3 f(x) 1 3x + = + + = + = ( )( )23 f(x) 1 0 x 223xf '(x) 03 f(x) 1+ = >+ x 0 f '(0) 0= . f , . ) x x 1= : ( ) ( )3 3f(1) f(1) 2 f(1) f(1) 2 0 ... f(1) 1+ = + = = ( )223 1 3f '(1)43 f(1) 1= =+. : 3 3 1y f(1) f '(1)(x 1) y 1 (x 1) y x4 4 4 = = = + )H f 1,1 .: f( ) f( 1),f(1) = f(1) 1= (1) x=-1 ( ) ( )( )( )2f( 1) 1 0 3 23f( 1) f( 1) ( 1) 1 f( 1) f( 1) 1 0 f( 1) 0 + > + = + + = = f( ) 0,1 = ) 3 f(x)2= f( ) 0,1 = 3 0 1 0 3 2 3 1 3 12 1 2 3= = =

mathhmagic.blogspot.gr : ., ., ., ., . 8165) f,g : , (1) g(x)f '(x)x= f(x)g '(x)x= x>0 : i) ( )f(x) g(x)h(x) ,x 0,x+= + . ii) f(1) 3= g(1) 1= ,: ) 22x 1f(x)x+= 22x 1g(x)x= ( )x 0, + ) f,g . i) ( )x 0, + : ( ) (1)2 2f '(x) g '(x) x f(x) g(x)f(x) g(x) xf '(x) xg '(x) f(x) g(x)h '(x) 'x x x+ + + = = = = 2f(x) g(x) f(x) g(x)0x+ = = . h . ii)) f(x) g(x)h(x) c c f(x) g(x) cxx+= = + = x>0. x 1= f(1) g(1) c 3 1 c c 4+ = + = = . , ( )x 0, + : f(x) g(x) 4x+ = g(x) xf '(x)= ( ) ( )2f(x) xf '(x) 4x xf(x) ' 2x '+ = = 21xf(x) 2x c= + x 0> x 1= 1 1 1f(1) 2 c 3 2 c c 1= + = + = 22 2x 1xf(x) 2x 1 f(x)x+= + = ( )x 0, + , f(x) g(x) 4x+ = g(x) 4x f(x)= 22x 1g(x) 4xx+= 22x 1g(x)x= ( )x 0, + ii) Cf,Cg 0 + 2x 0 x 02x 1lim f(x) limx+ + += = + 2x 0 x 02x 1lim g(x) limx+ + = = , x=0 Cf Cg. 22x xf(x) 2x 1lim lim 2x x+ ++= = ( )xlim f(x) 2x .. 0+ = = ,

mathhmagic.blogspot.gr : ., ., ., ., . 8222x xg(x) 2x 1lim lim 2x x+ += = ( )xlim g(x) 2x .. 0+ = = , y 2x= + Cf Cg. 66) (ll time classic) f , : 2xf(x) 1 121 f (x)+=+ (1) x f(0) 1 (2)= i) f. ii) f . iii) f . iv) Cf. v) f. vi) f . vii) 1f . viii) : 6 2 3x 1 9x 1 3x x+ + = i) 2 2 2 2 22xf(x) 1 12xf(x) 2 1 f (x) 1 f (x) 2xf(x) 1 f (x) 2xf(x) x x21 f (x)+= + = + = = + + ( ) ( )h(x) f(x) x2 22 2 2 2 21 x f (x) 2xf(x) x f(x) x 1 x h(x) 1 x (2)= + = + = + = + 21 x 0+ x h(x) 0 x . h . h(x) 0 x . h . h(0) f(0) 0 1 0= = > h(x) 0> x . ( )h(x) 02 2 2 2 2h(x) 1 x h(x) 1 x f(x) x 1 x f(x) x 1 x>= + = + = + = + + x . ii) ( )222 2 22x x 1 x xf '(x) x 1 x ' 1 12 1 x 1 x 1 x+ += + + = + = + =+ + + x x2 2 2 2 21 x x 1 x x 1 x x 1 x x 0>+ > + > + > + > x , 221 x xf '(x) 01 x+ += >+ x , f . iii) ( )22 2 21 x x 1f ''(x) ' ... 01 x 1 x 1 x + + = = = > + + + x . f . iv) ( ) ( ) ( ) ( )2 2 2 22x x x x x2 2 2x 1 x x 1 x x 1 x 1lim f(x) lim x 1 x lim lim limx 1 x x 1 x x 1 x + + + + = + + = = = = + + + x x x x x2 2 2 2 21 1 1 1 1 1lim lim lim lim limx1 1 1 1 1x x 1) x x 1) x(1 1) x(1 1) 1 1x x x x x + = = = = = + + + + + + + + +

mathhmagic.blogspot.gr : ., ., ., ., . 83x21 1 1lim 0 0x 211 1x = = + + . y=0 Cf 2 22x x x x1x(1 1)f(x) x 1 x 1xlim lim lim lim 1 1 2 x x x x+ + + + + + + + = = = + + = = ( ) ( ) ( )2 2x x xlim f(x) 2x lim x 1 x 2x lim 1 x x ... 0 + + + = + + = + = = = y 2x= Cf + . v) f ( ) ( )x xf(A) lim f(x), lim f(x) 0, += = + vi)H f 1-1 . iv) f-1 . ( )1fA f(A) 0, = = + . 22 y 1y f(x) y x 1 x ...... x2y= = + + = ( )21 x 1f (x) ,x 0,2x = + . vii) ( )21 2x 0 x 0 x 0 x 0x 1 1lim f (x) lim lim x 1 lim2x 2x+ + + + = = = x=0 Cf-1 1 22x xf (x) x 1 1lim lim x 22x+ += = = 21x x1 x 1 xlim f (x) x lim ... 0 2 2x 2+ + = = = = 1y x2= Cf-1 + . viii) ( ) ( )2 26 2 3 6 3 2 3 3x 1 9x 1 3x x x 1 x 9x 1 3x x 1 x 3x 1 3x+ + = + + = + + + + = + + ( )3 3f(x ) f 3x x 3x x 0 x 3 x 3= = = = =

mathhmagic.blogspot.gr : ., ., ., ., . 84 (Rolle ) 67) , , , ,, . f(t) t, f(t) . g(t) t, g(t) . t=0 , f(0) g(0) , f() g()= = . , , 0t , 0 0f '(t ) g '(t )= . h(t) f(t) g(t)= ( )0, h 0, h ( )0, h(0) f(0) g(0) 0 , h() f() g() 0= = = = Rolle , , , ( )0t 0, 0 h'(t ) 0 f '(t ) g '(t ) 0 f '(t ) g '(t )= = = , . t=0 t=

mathhmagic.blogspot.gr : ., ., ., ., . 8568) f : 0,2 f(x) x= x 0,2 i) f . ii) 1f (0) . iii) 1f , 1x 01974limf (x) x iv) : 2f(x) x x 0,2 i) f(x) x= 0,2 = , f f( ) f(0),f( ) 0,12 = = ii) ( )1 1 1f(0) 0 f f(0) f (0) 0 f (0) = = = iii)x 0,2 x 0+ . x 02> > x x f(x) x< < Cf y=x 1Cf y=x 1 1f (x) x f (x) x 0 > > x 0,2 ( )1x 0lim f (x) x 0 = 1x 01974limf (x) x= +. iv) 2x x (1 ) x 0,2 x 0,x2= = (1) x 0,2 x2 2x x x xg(x)x= ,x 0,2 . ( ) ( ) ( )2 2x ' x x x' x x xxg '(x) 'x x x = = = (1) , h(x) xx x= ,x 0,2 ( )h'(x) xx x ' x xx x xx 0= = = < x 0,2 h 0,2 . x 02 > : h(x) h(0) xx x 00 0 xx x 0< < < (2) (1),(2) g '(x) 0< , g x 0,2

mathhmagic.blogspot.gr : ., ., ., ., . 86 x 0,2 :x x x 1 2 22g(x) g( ) x x 2 x x x 2 2 (Bonus ) x x< , x 0> . g(x) x x= x 0 .: g '(x) x 1 0= ( 1 x 1 2 x 1 0 ) x 0 , g )0, + , x 0 g(x) g(0) x x 0 x x> < < < 69) ( )f : 0,+ : f ( )0,+ f(1) 1= Cf 0 0M(x ,f(x )) x'x 1A(x ,0) 10xx2= . f. i) Cf 0 0M(x ,f(x )) (): 0 0 0y f(x ) f '(x )(x x ) = H () 1A(x ,0) 10 0 1xx 2x x20 0 1 0 0 0 1 0 0 0 0 00 f(x ) f '(x )(x x ) f(x ) f '(x )(x x ) f(x ) f '(x )(2x x )= = = = = 0 0 0 0 0 0f(x ) f '(x )x x f '(x ) f(x ) 0 = = , ( )0x 0, + ( )x 0, + : ( )x 0 cxf '(x) f(x) 0 xf(x) ' 0 xf(x) c f(x)x> = = = = , c , ( )x 0, + f(1) 1 .... c 1= = . 1f(x) ,x= ( )x 0, + 70) f,g f(x) g(x) x 4 = x . y 3x 7= f x + . ) i) xg(x)limx+ ii) 2xg(x) 3x 2xlimxf(x) 3x 1++ + + ) y 2x 3= g x + . )i) y 3x 7= f x + xf(x)lim 3x+= (1) + x 0> , f(x) g(x) g(x) g(x)x 4 f(x) 4 f(x) 4f(x) g(x) x 4 1 1x x x x x x x x = = = = + ( ) (1)x x x xg(x) g(x)f(x) 4lim lim 1 lim 3 1 0 2 lim 2x x x x+ + + += + = + = =

mathhmagic.blogspot.gr : ., ., ., ., . 87ii) 2x xg(x) 2x3g