λυσεις 1 50
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1. 1 1 1 x f(x) ln x = ) ( ) ( ) 1 x 0 1 x x 0 x 0,1 A x > > = , ( ) ( ) 1 f (x) 0 , f 1 x x = < 2 ) x 0 1 x u x 1 x ux 0 x 0 lim x 1 x lim f(x) lim ln lim ln u x+ + + = + =+ = = = + , x 1 1 x u x 1 xx 1 x 1 u 0 lim 0 x 1 x lim f(x) lim ln lim ln u x + = = = = = , ( )f(A) ,= + ) f ''1 1'' , 1 x f(x) y y ln x = = ( )y 1 1 y x 1 x 1 1 e x , f : 0,1 f (x) x 1 e 1 e = = = + + r ) x x x1 1 1 1 11 x x x x 00 0 0 0 1 1 e e (1 e ) 2e E f (x) dx dx dx 1 dx x ln(e 1) ln 1 e 1 e 1 e e 1 + + = = = = = + = + + + + 2 ln x f(x) x 2 x = ) ( ) ( ) 2 1 1 2 x ln x 1 2x 2 ln xx xx 0, A ,f (x) , 2 x 4x x2 x + + = = = 4x x 0> , ( ) 1 g(x) 2x 2 ln x ,x 0 , g (x) 2 0 ,g , x 1 g(x) g(1 ) 2x 2 ln x 0 x = + > = + > > > + >1 , 0 x 1 g(x) g(1) 2x 2 ln x 0< < < + < ( ) , f x 1= f(1) 1= ) f(x) 1 f( 1) f(e ) 2+ + = f(e ) 1 e 1 0 f( 1) 1 1 1 = = = + = + = ) 0 1< < ( ) ( ) 1 1 1 11 1 ln x 2 2 1 () f(x) dx x dx x x x ln xdx 1 x ln x dx 3 32 x x = = = = + = 2 2 2 8 ln 2 2 ln 2 3 3 3 3 = + + + = + + (1) , 1> 1 () f (x) dx= = 1 (1) 1 f(x)dx f(x)dx= = 2 8 ln 2 3 3 + ) 0 0 0 2 8 8 lim() lim () lim ln 2 3 3 3+ + = = + + = , 0 2 8 8 lim 2 3 3 3+ + = 2. 2 ( ) ( ) ( ) ( )DLH 0 0 0 0 0 1 lnln lim ln lim lim lim lim 2 0 1 1 1 2 + + + + + + = = = = = 2 8 2 ln 2 8 lim() lim ln 2 lim 3 3 3 3 + + + = + = + = + , 2 2 8 2 lim 0 , lim 3 33 + + + = > = + ( ) ( ) ( ) DLH lnln 1 lim lim lim 0 + + + + = = = = 3 x f(x) e x 1 = + ) x r , x x x x x x 1 e 1 f (x) e 1 1 , e 0,f (x) 0 e 1 0 x 0, e e = + = + = > > > > f (x) 0 x 0 < < f x 0= f(0) 0= , ( )f x 0 f(x) f(0) f(x) 0< > > 2 , ( )f x 0 f(x) f(0) f(x) 0> > > 1 , f ) ( )x x x xx x x x x 1 1 1 lim f(x) lim x 1 , lim f(x) lim x 1 lim 1 xe 1 e e e+ + = + = + = + = + = + , ( ) ( ) x x x x xx x x DLH x x 1 x (x) 1 lim , lim xe lim lim lim 0 e e (e ) e + = + = = = = x x lim (1 xe ) 1 0 + = > 0 , [ )f(A) 0,= + , x (e 0) x x x x e (x 1)e 1 x 1 e e x 1 f(x) > = = + = = , ( ) [ )1 2 ,0 , 0,= = + ( ) [ )1 2f(A ) 0, , f(A ) 0,= + = + 0< f(A) 0= x 0= 0> 1 2 f(A ) , f (A ) 1 2A ,A f ) [ ) 2 x xx g(x) 1 x e , x 0, , g (x) 1 x e f(x) 0 2 = + + = + + = > ) , ( ) 2 2g x xx x x 0 g(x) g(0) 1 x e 0 1 x e 2 2 > > + > + > 1 ) ( )2 f(x) f x ln x= + x 1= ( )2 f(x) f x ln x = ( )f 2 2 2 x 1 ln x 0 , x x 1 f (x ) f(x) f(x) f(x ) 0> > > > > < 1 , 3. 3 ( )f 2 2 2 0 x 1 ln x 0 , 0 x x f(x ) f(x) f(x) f(x ) 0< < < < < < > 1 , , x 1= 4 f [ ], , ( ), f() , f () = = ) [ ]g(x) f(x) x , g() f() 0 , g() f() 0, ,= = = = = . Rolle , ( ) , g () 0 f () 1 0 f () 1 = = = ) ( ), 1 2 d , d = = 1 2d d 2 1 3 = = , ( )1 2 d 3 = ( ) 2 2 3 3 + = + = f ... [ ] [ ], , ( )1 , ( )2 , ( ) ( )1 2 f () f() f() f() f f = = , 1 2 f() f () 2f ( ) f ( ) 2 2 3 + = + + ( )3 f() f () f() f ()f() f() f() f() f() f() 3( ) 2 3 2 2 3 3 3 + + = + = = = + ) 2 2 2 h(x) f(x) , h() f() 0 h() 0 3 3 3 3 3 + + + = = = = < = > h() h() 0 < h . Bolzano ( )0 0x, h(x ) 0 = 0 2 f(x ) 3 + = ( ... 2 f() f() 3 + = < < = ) ) f ... [ ] [ ]0 0,x x , ( )1 0x,x ( )2 0 x x , ( ) ( )0 0 1 2 0 0 f(x ) f() f() f(x ) f x f x x x = = , ( ) ( ) ( )0 0 0 00 0 1 2 3 x 3 x 3 x xx x1 2 2 2 3 2 2 f (x ) f (x ) 2( ) 3 3 + + = + = + = = + + 5 ( )f : 0, f(1) 1+ =r , 3 2 x f (x) 2x x 2 + = + x 0> ) 3 2 3 2 3 2 2 1 2 2 1 2 x f (x) 2x x 2 x f (x) x 2 2x f (x) f (x) ln x x x x x x + = + = + = + = + 2 2 1 2 1 2 f(x) ln x c , f(1) 1 c 0 f(x) ln x , x 0 x x x x = + + = = = + > ) ( ) 2 3 x 2x 2 f (x) 0 x 0 , f 0, x + = > > + 1 ( )2 2 2x x x 0 x 0 x 0 1 2 1 2 1 lim f(x) lim ln x , lim f(x) lim ln x lim ln x 2x 1 x x x x x+ + ++ + = + = + = + = + = 2 x 0 x 0 x 0 1 lim ln x , lim(2x 1) 1 0 , lim x+ + + = = < = + f(A) = r , 4. 4 2 1 2 2015 2 2 1 2 1 2 e ln 2015 ln 2015 f() 2015 + = = + + = = , 2015 f(A) = r ( )f 1 ''1 1'' ( )0, + f() 2015= ) 2x 0 x 2 , 2 x 0 x 2 > < > < , ( ) ( ) ( )f f 2x f(2 x) 0 f 2 x f(2 x) 2 x 2 x x x > > > < 1 , x x ,x 0 x x x< < < , x 0 , xx x> < < ( )x 0,2 x 0 , xx x< < < , 0 x 2< < ) f 1 ,1 2 1 f(1) 1 0 f ln 2 0 2 = > = < , . Bolzano 0 0 1 x ,1 f(x ) 0 2 = ( )f 1 ( )+,0 x x x x x 2 2 1 1 1 1 2 1 g (x) e ln x , g (x) e ln x e e ln x e f(x) x x x x x x = + = + + = + = , ( )x e 0 x 0g (x) 0 e f(x) 0 f(x) 0 x x > = = = = , ( )f 0 0x x f(x) f(x ) f (x) 0 g (x) 0> > > > 1 , ( )f 0 00 x x f(x) f(x ) f(x) 0 g (x) 0< < < < < 1 g 0x x= 6 2 x x 1 f(x) x 1 + + = ) ( )2 2 0 2 2x x x f(x)x x 1 x lim lim lim 1 x x x x + + + + + = = = = ( ) ( )2 2 x x x 1 x 1xx 1 x x lim f(x) x lim lim 1 0 1 x 1 x 1+ + + + ++ + + = = = + = = ) ( ) ( )( ) ( ) ( ) 2 2 2 2 2 x x 1 (x 1) x x 1 x 1 x 2x f (x) x 1 x 1 + + = = = L , ( ) 3 2 f (x) x 1 = = L , f f x 0= f(0) 1= x 2= f(2) 3= ) ( )x 1, + f(x) f (0) 3 = , 2 4 x 1 , x 1 1+ + > x 0 , ( ) ( )2 4 f x 1 f x 1 6+ + + = ( ) ( ) 2 2 2 4 4 4 x 1 2 x 1 f x 1 f x 1 3 x 1 2 x 1 + = = + = + = + = = x 1 x 1 = = 5. 5 ) 20 0 0 0 0 1 1 1 1 1 x x 1 1 1 E f (x) x dx x dx dx dx ln x 1 ln 2 x 1 x 1 x 1 + = = = = = = 7 x f(x) e x 1= + ) ( )x f (x) e 1 0 , f ''1 1'' = + > 1 r x x lim f(x) , lim f(x) + = = + ( )1 f D f(A) , = = + = r ) : ( )1 1 y f (e) f (e) (x e) = ( )f ''1 1'' 1 f (e) f() e f(1) 1 = = = = . 1 f (e) 1 = ( )1 f f (x) x , x = r , 1 f ( ) ( )1 1 f f (x) f (x) 1 (1) = ( ) ( ) ( ) ( ) ( )1 1 1 1 1 x e , f f (e) f (e) 1 f 1 f (e) 1 f (e) e 1 = = = = + , : 1 1 1 y 1 (x e) y x e 1 e 1 e 1 = = + + + + ) ) x x f (x) e 1 0 , f (x) e 0 = + > = > , (1) ( )1 f f (x) 0 > ( ) ( ) 1 1 1 f (x) f f (x) = f ( )1 f ( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 1 1 2 2 1 1 f f (x) f f (x) f (x) f (x) 0 f f (x) f f (x) = = < ( )1 f (x), f (x), f (x) 0 > , 1 f ) : y 0 11 1 y x x 1 , f(0) 0 f (0) 0 x 0 e 1 e 1 = = + = = = = + + f ''1 1'' , 1 f ( )e 1 1 0 1 e 2 E E (e 1) 1 f (x)dx 2 2 = = + = ( ) 1 1 1 1 1 1 2 2 2u f (x) f(u) x , dx f (u)du ,u f (0) f(u ) 0 u 0,u f (e) f(u ) e u 1 = = = = = = = = = [ ] 12e 1 1 11 u 0 0 0 0 0 u 3 f (x)dx uf (u)du uf (u) f(u)du f(1) f(0) e u 2 2 = = = + = = K 6. 6 8 f : r r f(0) 0= 1 1 f (x) 2x , x 0 x x + = ) ( ) ( ) ( )2 2 21 1 1 1 f (x) 2x x x x x , x ,0 0, x x x x = = + = + 2 2 1 x c , x 0 x 1 f(x) x k , x 0 x 0 , x 0 + < = + > = , f 0 x 0 x 0 lim f(x) lim f(x) f(0) 0 (1)+ = = = 2 2 2 2 2 21 1 1 x x x x x x x x x = 2 x 0 1 lim x 0 x = (1) c k 0= = 2 1 x , x 0 f(x) x 0 , x 0 = = , x 0 x 0 f(x) f(0) 1 f (0) lim lim x 0 x 0 x = = = , : y f(0) f (0)(x 0) y 0 (x x) = = ) 1 1 1 1 f(x) 0 x 0 0 , x , x 0, 0 x x = = = = = > Z 1 (0,0) ,0 1,2,3, = = K ) 1 u 0 x x x u 0 f(x) 1u lim lim x lim 1 x x u+ = > + + = = = ( ) ( ) 1 0 u 0 x 0 2 2 2 2x x DLHu 0 u 0 u 0 u 0 1u 1 u u (u u) 1 u lim f(x) x lim x x lim lim lim lim 0 x u u u (u ) 2u+ + + + = > + + = = = = = = y x= + ) ( ) 21 1 1 1 4 4 21 1 1 1 2 2 2 2 1 1 x f(x) x xE g(x) dx dx dx dx x x x = = = = 2 1 1 u , du x x = = [ ] 2 2 Eudu u 2 2= = = + = ( ) 1 1 1 1 1 x , x , 0 f(x) 0 2 2 x 9 ( )2x x1 f(x) 4e 1 x 1 , x , g(x) x , x h(x) e 2 = = =r 7. 7 ) ( )2x 2x 2x 2x f (x) 4 2e (1 x) e ( 1) 4e (1 2x) , 4e 0 = + = > f 1 x 2 = 1 f 2e 1 2 = ) ( ) ( )x 2x x 1 e 1 x 4e 1 x 1 f(x) 0 4e = = = ( ) ( ) 2x 2x 2x 2x 2xx x DLH x x x 1 x (1 x) 1 1 lim e 1 x lim lim lim lim e 0 e (e ) 2e 2 + + = = = = = x lim f(x) 1 = , 1 x 2 lim f(x) 2e 1 = , x lim f(x) + = 1 2 1 1 A , , A , 2 2 = = + ( ) ( )1f A 1,2e 1= ( ) ( ]2f A ,2e 1= ( ) ( )1 20 f A ,0 f A f , ) ( ) 4e 1 e f() 0 = = ) 2 1 A , 2 = + 1 f 2e 1 0 , f(1) 1 0 2 = > = < 1 ,1 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g hg 1 1 1 , 1 2 1 1 1 2 2 2h 2 2 2 1 2 : y g(x ) g x x x g x h x 1 g(x ) x g x h(x ) x h x : y h(x ) h x x x , x ,x 2 = = = = ( ) ( ) ( ) 2 2 2 2 2 2 x 1 x x1 2 2 2x x x 1 1 2 1 1 e 2 x x 1 e 1 x e 1 x f x 0 1 2 4e x x e x e 2 x = = = = + = + ) 1 ,1 2 10 2 ln x x f(x) 2 x 2 = + ) 3 2 2 1 ln x 1 ln x x f (x) x , x 0 x x = = > , ( ) ( )3 21 g(x) 1 ln x x , x 0 , g (x) 3x 0 , g 0, , g(1) 0 x = > = < + =2 ( )g 3 x 1 g(x) g(1) 1 ln x x 0 f (x) 0> < < < 2 , ( )g 3 0 x 1 g(x) g(1) 1 ln x x 0 f (x) 0< < > > > 2 f x 1= 3 f(1) 2 = 8. 8 ) ( ) ( )3 3 x 2 42ln x x 2ln x x x 2 4 f(x) 2x 2x 2x = + = + = ( ) ( )x 0 x 0 x 0 ln x 1 lim f(x) , lim lim ln x x x+ + + = = = + = , x lim f(x) + = ( )x DLH x x ln x (ln x) 1 lim lim lim 0 x (x) x + + + + + = = = 2 x x lim 2 2+ + = , ( ) [ )1 2 0,1 , 1,= = + 1 2 3 3 f(A ) , , f(A ) , 2 2 = = 3 2 > 1 2 f(A ),f(A ) 3 2 = x 1= 3 2 < ( ) ( )1 2x 0,1 ,x 1, + 1 2 f(A ) , f (A ) 1 2A ,A f ) ( ) t2 2 3t t t t 2 1 1 1 1 1 ln x x x 1 x f(x)dx 2 dx ln x ln xdx 2 dx ln x 2x x 2 2 2 6 = + = = + = 3 21 t 11 ln t 2t 2 6 6 = + , ( ) 3 2t 2 3 3 2 3t t t1 1 t 11 ln t 1 2 11 1 lim f(x)dx lim ln t 2t lim t 2 6 6 2t 6 t 6t 6+ + + = + = + = + = ( ) ( ) ( ) ( ) 22 3 3 2 3 3t DLH t t t DLH t ln t ln t ln tln t ln t 1 lim lim lim lim lim 0 2t (2t ) 3t 3t 9t + + + + + + + + + = = = = = ) 3 f 2 = 3 3 f() , f() f() f () 3 2 2 + , 2 3 3+ 2 2 3 f() f() 3 f() f() 1 1 02 3 3 0 + = = = = = = + = = 11 1 f(x) x ln 1 ln x x = + + ) 1 1 1 1 1 1 1 f (x) ln 1 x 1 ln 1 0 1x x x x x x 11 x = + + + + = + + > + + x 0> , 1 1 1 1 1 1 1 1 ln 1 0 , 0 x x 1 0 x x x x 1 x x 1 + > + > < < + > > + + f ( )1 ) ( )0,+ ''1 1'' ( ) x 0 1 u 1 x u DLH u1x 0 lim 1 x 1 ln u 1 lim x ln 1 lim lim 0 x u 1 u+ + + = + + + + + =+ + = = = x 0 lim ln x+ = x 0 lim f(x)+ = , 9. 9 ( ) x 1 0 u 1 x 0 x u 1 DLH u 11 lim 1 1 x 1 ln u 1 lim xln 1 lim lim 1 x u 1 u + = + + + = + = = = x lim ln x + = + x lim f(x) + = + , ( )f(A) ,= + = r ) x x 1 1 ln ln x 2015 xln 1 ln x 2015 f(x) 2015, x 0 x x + + = + + = = > 2015 f(A) f ''1 1'' ) ( ) ( )1 1 1 ff f : D f(A) , f (A) D 0, = = + = = = + r , 1 f f , ( )1 1 1 x x f (A) lim f (x), lim f (x) + = 1 x lim f (x) + = + 1 u f (x) f(u) x = = ( ) ( )1 x u DLH u u x f (x) f(u) u f (u) 1 1 lim lim lim lim 1 x 1 f(u) 1 f (u) f (u) + + + + + + + + + = = = + =+ ( ) u u u 1 1 1 1 lim f (u) lim ln 1 ln(1 0) 0 0 0 , f (u) 0 lim u u u 1 f (u)+ + + = + + = + + = > = + + ) ( ) ( ) ( ) 2x 12 2 2 2 f (x) 01 1 1 1 1 x 1 E f(x) dx x ln 1 ln x dx ln 1 dx x ln x x 2 x > > = = + + = + + = [ ] 2 22 2 2 22 2 1 1 1 1 1 1 1 x x 1 1 3 1 x x ln 1 dx x ln x dx 4ln ln 2 dx 2ln 2 1 2 x 2 x 1 x 2 2 2 x 1 = + + = + + = + + ( ) [ ] 2 1 1 1 3 1 4ln3 5ln 2 x ln(x 1) 2ln 2 1 ln3 2 2 2 2 = + + + = = K 12 f : r r x 1 f(x) 3x lim 2 f(2) 3 x 1 = = ) f ( )f 2 , f x 1 f(1) limf(x) 3 = = ( ) x 1 x 1 x 1 3 x 1f(x) 3x f(x) 3 3 3x f(x) f (1) lim 2 lim 2 lim 2 f (1) 3 2 f (1) 5 x 1 x 1 x 1 x 1 + = = = = = fC (1,f(1)) : y f(1) f (1)(x 1) y 3 5(x 1) y 5x 2 = = = f fC . f(x) 5x 2 2 5x f(x) 0 + '' ''= x 1= ( ) ) f(1) f(2) 3= = . Rolle ( )0 0x 1,2 f (x ) 0 = ( )f 2 ''1 1'' 0x , (f ) 0 0x x f (x) f (x ) f (x) 0 < > > 2 (f ) 0 0x x f (x) f (x ) f (x) 0 < > > 2 f 0x x= 10. 10 ) x 0 F(x) f(t)dt= 1 F (x) f(x) 0,x , 5 = < ) 1 x 5 = 1 f 1 5 ) 1 f f 1 0 5 0 1 1 x 1 x f(x) f f(x) 0 5 5 < < < < < < 1 , ( ) 1 F , ''1 1'' 5 2 2x 0 2x F''1 1'' x x 0 f(t)dt 0 f(t)dt f(t)dt 0 F(2x) F(x) 2x x x 0 = + = = = = : , f(x) 0< 2x 2x x x x 2x x 2x f(t)dt 0 f(t)dt 0< > < > , 2x x f(t)dt 0= , 2x x x 0= = ) ) 2 x 5 = 2 f 0 5 ) 2 f f 0 5 0 2 2 x 1 x f(x) f f(x) 0 5 5 < < < < < 1 , x 2 2 0 g (x) 2x f(t)dt x f(x) 2xF(x) x f(x) = + = + , 2 2 x 0 , f(x) 0 x f(x) 0 < , 2xF(x) 0 2 x 0 0 x 5 < ... F [ ] [ ]0,x x,0 ( ) ( ) 0,x x,0 F(x) F(0) F () F(x) xf() x 0 = = 2 2 2F(x) 2x f() 0 f() 0 x 0= < , 2 g (x) 0 ,x , 5 '' ''= x 0= g 2 , 5 13 ( )f : 0,+ r x 1 1 f(t) f(x) 2x 1 dt , x 0 x t = > ) ( ) ( )2 2 1 f(x) 1 f (x) 2 xf (x) 2x f(x) x f(x) xf (x) x ln x x x x = + = + + = + ( ) ( )2 2 ln x 1 xf(x) x ln x xf(x) x ln x c , f(1) 0 c 1 f(x) x , x 0 x = + = + + = = = + > ) 2 2 x 2 ln x f (x) 0, x 0 x + = > > ( )2 g(x) x 2 ln x , x 0,= + + , 2 2x 1 g (x) x = g 2 x 2 = 2 5 ln 2 g 0 2 2 + = > 11. 11 2 g(x) g 0 2 > , ( ) ( )f 0, + 1 f(1) 0= ( )f x 1 f(x) f(1) 0> > = 1 , ( )f 0 x 1 f(x) f(1) 0< < < = 1 f ( ) ( ) ( )x 0 x 0 ln x 1 1 lim lim ln x 1 x x+ + = = + = x 0 lim f(x)+ = x 0= ( )y y 0 ( )2 2 2x DLH x x ln x 1 (ln x 1) 1 lim lim lim 0 x (x ) 2x + + + + + = = = 2x x f(x) ln x 1 lim lim 1 1 0 1 x x+ + = + = + = ( ) ( )x x DLH x x ln x 1 (ln x 1) 1 lim f(x) x lim lim lim 0 x (x) x + + + + + + = = = = y x= + ) ( ) ( ) 121 1 1 1 ln x 1 1 ln x ln x E() f(x) x dx dx dx ln x 1 ln x dx ln x x x 2 = = = = = = 2 ln ln 2 = + ) ( ) ( )2 f 2 f ''1 1'' z ln z 1 z ln z 1 0 f z f(1) z 1 z + + = = = = 1 , z (0,0) 1 14 [ )f : 0,+ r x f(0) 1 2x f (x) e , x 0= < < > ) ( )f (x) 2x 0 , x 0, > > + f 0, f ( ) [ ) 0,+ 1 ( )2 f (x) 2x 0 f(x) x 0 > > . ( ) [ )2 g(x) f(x) x 0,= + 1 g 2 2 x 0 g(x) g(0) f(x) x f(0) f (x) x 1> > > > + 1 ( )x x f (x) e 0 f(x) e 0 < < . ( ) [ )x h(x) f(x) e 0,= + 2 h x x x 0 h(x) h(0) f(x) e f(0) 1 f(x) e> < < < 2 ) ( ) 2 x 2 x x 0 x 0 x 0 x 1 f(x) e lim f(x) 1 f(0) lim x 1 1 lim e + + + + < < = = + = = , 2 2 1 1 0 x 1 f (x) 0 x 1 f(x) < + < > > + 2x 1 lim 0 x 1+ = + x 1 lim 0 f(x)+ = f (x) 0 x x 1 lim f(x) lim 1 f(x) > + + = = + [ )f(A) 1,= + ) [ ]2 (x) f(x) 2x , x 1,2= , ) x 1 x 2= = 2 f(1) e< < 2 5 f(2) e 8< < < (1) f(1) 2 0 , (2) f(2) 8 0= > = < , . Bolzano ( )0x 1,2 2 0 0 0(x ) 0 f(x ) 2x= = ) 1 1f (x) 0 0 0 E() f(x) dx f(x)dx > = = , 2 x x 1 f(x) e+ < < 12. 12 ( ) 131 1 1 12 x x 00 0 0 0 x 4 x 1 dx f(x)dx e dx x E() e E() e 1 e 3 3 + < < + < < < < < 15 ( )f : 0,+ r f(1) e , f (1) 0= = 1 x xf (x) f(x) e x 0 = > ) ( ) ( ) 1 1 1 1x x x x 2 2 f (x)x x f(x) e f(x) xf (x) f(x) e f (x)x x f(x) e e x x x = = = = 1 1 1x 1 x x x f(x) f(x) e c f(1) e c e e c c 0 , e f(x) xe x x = = + = + = + = = = ) 1 1 1 1 1 x x x x x 1 1 x 1 f (x) e xe e 1 e ,x 0 ,e 0 x x x = + = = > > x 1= f f(1) e= , f(x) f (1) x x1 1 x x xx x x 1 xe e 1 xe e x e e e e > ) ( ) 1 u1 u ux ux u DLH u ux 0 x 0 e (e ) lim f(x) lim xe lim lim lim e u (u)+ + + = + + + + = = = = = + , x 0= ( )y y 0 1 u1 x ux x x u 0 f(x) lim lim e lim e 1 x + = + + = = = , ( ) ( ) 01 u1 u u 0x x x x DLHu 0 u 0 e 1 e 1 lim f(x) x lim xe x lim lim u u u+ + = + + = = = = ( ) ( ) u u u 0 u 0 e 1 lim lim e 1 u + + = = = , y x 1= + + 1 1 x x 1 f(x) x 1 xe x 1 e 1 , x 0 x > + > + > + > , [ )x g(x) e x 1 , x 0, ,= + x g (x) e 1 0 = > , ( ) [ )g 0, + 1 g x x x 0 g(x) g(0) e x 1 0 e x 1 , x 0> > > > + > 1 1 0 x > 1 x 1 e 1 x > + ) 2 2 1 1 E f(x) x 1 dx (f(x) x 1)dx= = ) f(x) x 1> + f 1 x 2 f(1) f(x) f(2) e f (x) 2 e e x 1 f(x) x 1 2 e x 1 1 '' ''= x 1 , x 2= = , ( ) ( ) ( ) ( ) ( ) 2 22 22 2 2 1 1 1 1 1 x x e x 1 dx f(x) x 1 dx 2 e x 1 dx e 1 x E 2 e 1 x 2 2 < < + < < + ( ) ( )1 1 5 5 2 (e 1)2 (e 1) E 2 2 e 1 2 2 e 1 e E 2 e 2 2 2 2 + + < < + + < < 16 13. 13 f : r r f(0) 0= , f (x) f (x) f (x) x e x , f (x)e 1 x e + = + x r ) f (x) g(x) e x 0= + g , , f (0) g(0) e 0 1 0= + = > , g(x) 0> x r ( ) 2 f (x) f (x) 2 f (x) x x 1 x f (x)e 1 x e g (x)g(x) x g (x) x e g(x) 2 2 + = + = = = + x 0 2 2 21 1 1 1 g (x) x c g (0) c c 2 2 2 2 = = + = = g(x) 0 2 2 2 21 1 1 g (x) x g (x) x 1 2 2 2 > = + = + ( )2 f (x) 2 f (x) 2 2 g(x) x 1 e x x 1 e x 1 x f(x) ln x 1 x = + + = + = + = + 2 2 2 2 2 2 2 1 2x 1 x x 1 1 f (x) 1 f (x) x 1 1 0 , x x 1 x 2 x 1 x 1 x x 1 x 1 + = = = + + = + + + + + r ) ( )2 1 f (x) 0 , f x 1 = < + 2 r ''1 1'' ( ) x 0 2 2 2 2x x x 1 1 lim x 1 x lim x 1 x lim x 1 1 x x < + = + = + = + , ( )( ) 2 u x 1 x 2 x x u lim f(x) lim ln x 1 x lim ln u = + + = + = = + , ( ) ( ) ( ) ( ) 2 2 2 x x x2 2 x 1 x x 1 x 1 lim x 1 x lim lim 0 1x 1 x x 1 1 x + + + + + + + = = = + + + + , ( )( ) 2 u x 1 x 0 2 x x u 0 lim f(x) lim ln x 1 x lim ln u+ = + > + + = + = = ( )f(A) ,= + = r 2y y y y 2 y 2 2y 2 y 2 y 1 e e e y f(x) e x 1 x e x x 1 e x 2xe x 1 x 2e 2 = = + + = + + + = + = = 1 f : r r x x 1 e e f (x) 2 = ) 1 x x f (x) e e h(x) , x 2x = = 1 x x x x f ( x) e e e e h( x) h(x), ( x) 2x 2x = = = = . h 1 1ln ln 2 2 ln2 ln 2 f (x) dx h(x)dx 0 x = = : h h(x)dx 0 = ) f 0 x 1 f(0) f(x) 0 f(x) 2 ( ) ( ) [ ] 1 1 1 1 1 12 0 0 0 0 0 0 E f(x) dx f(x)dx ln x 1 x dx x f(x)dx xf(x) xf (x)dx = = = + = = + = ( ) ( ) ( ) ( ) 21 1 1 2 2 2 00 0 x 1x f(1) dx ln 2 1 dx ln 2 1 x 1 ln 2 1 1 2 x 1 2 x 1 + = + = = + = + + + 14. 14 17 [ ]f,g : 2,3 r [ ]2,3 ( )2,3 f (x) 0 ( )x 2,3 ( ) ( )2 2x 5 g(x) x 5x 6 g (x) + (1) [ ]x 2,3 ) f (x) 0 f ''1 1'' , ( )1 2x ,x 2,3 1 2 1 2x x f(x ) f(x ) = . Rolle ( ) ( ) ( )0 1 2 2 1x x ,x x ,x 2,3 0f (x ) 0 = , h(x) 5f(x) 2f(2) 3f(3)= , ( ) ( )h(2) 3 f(2) f(3) , h(3) 2 f(2) f(3)= = ( ) 2 h(2) h(3) 6 f (2) f(3) 0 = < f(2) f(3) . Bolzano ( ) 2,3 2f (2) 3f(3) h() 0 5f() 2f(2) 3f(3) f() 5 + = = + = , f ''1 1'' ) ... ( ) ( )1 2 2, , ,3 1 2f (2) 3f(3) f(2) f() f(2) 5f ( ) 2 2 + = = = ( ) ( ) 3 f(3) f(2) 5 2 = ( ) ( )2 2f(2) 3f(3) f (3) 2 f(3) f(2)f(3) f() 5f ( ) 3 3 5 3 + = = = ( ) ( ) ( ) ( ) 2 1 2 6 f(3) f(2) f f 0 25 2 (3 ) = > 2 0 2 3 3 0 > < < < ) (1) x 2= g(2) 0 g (2) g(2) 0 x 3= g(3) 0 g (3) g(3) 0 g(x) 0 ( )x 2,3 2 x 5x 6 (x) g(x) + = [ ]2,3 . Rolle, ( )0x 2,3 ( ) ( ) ( ) ( ) ( ) 2 0 0 0 0 0 2 0 0 0 0 0 02 0 2x 5 g(x ) x 5x 6 g (x ) x 0 0 2x 5 g(x ) x 5x 6 g (x ) g (x ) + = = = + , (1), g(x) 0 ( )x 2,3 . ( ) 2,3 g() 0= ) g (x) 0 > ( )x 2,3 , g ... [ ]2,3 ( ) ,3 ( ) ( ) g(3) g() g g(3) g (3 ) 3 = = (2) ( ) ( ) g () 0 (2) 3 3 3 3 0 g ()(3 ) g 3 g(3) g )(3 ) > < < > > > > > > 18 ( )f : 1,1 r f (0) 2 = ( ) x y f(x) f(y) f , x, y 1,1 1 xy + + = + ) y 0= x 0 x 0 f(x) f(0) f(x) f(x) f(0) f(x) f(0) 0 , 2 f (0) lim lim x 0 x + = = = = = x ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 x y 1 xy x y 1 xyx y x y 1 y f (x) f f 1 xy 1 xy1 xy 1 xy + + + + + + = = + ++ + x 0= 15. 15 ( )2 2 2 f (0) 1 y f (y) f (y) y 1 = = , x x 2 0 0 2 f (y)dy dy y 1 = [ ] ( )x 1,1x xx 0 0 0 1 1 f(y) dy f(x) f(0) ln y 1 ln y 1 f(x) ln x 1 ln x 1 y 1 y 1 = = + = + + 1 x f(x) ln(1 x) ln(x 1) f(x) ln 1 x = + = + ) ( ) ( ) ( )2 2 f (x) 0 , x 1,1 , f 1,1 x 1 = < 2 ''1 1'' ( ) y y y y y y y 1 x 1 x 1 e f(x) y ln y e 1 x e xe 1 e 1 e x x 1 x 1 x 1 e = = = = + = + = + + + 1 x u 1 x ux 1 x 1 1 x lim f(x) lim ln lim ln u 1 x+ + = + + = = = + + , 1 x u 1 x x 1 x 1 u 0 1 x lim f(x) lim ln lim ln u 1 x + = + = = = + , f(A) = r ( ) x 1 1 x 1 e f : 1,1 f (x) 1 e = + r ) x1 1 1 x 0 0 e 1 E f (x) dx dx e 1 = = + , x x du e u , e dx du udx du dx u = = = = , x 0 u 1 , x 1 u e= = = = ( )e e e 1 1 1 u 1 du A B 1 2 E du du u 1 u u u 1 u u 1 = = + = + = + + + [ ] ( ) ( ) ( ) e2 2 2 e 1 1 u 1 e 1 e 1 ln u 2ln(u 1) ln ln ln 4 ln u e 4e + + + = + + = = = ( ) ( ) A B 1 B 2A B u Au 1 A B A 1 A 1u(u 1) u u 1 u(u 1) + = =+ + = + = = = + + + ) ( ) 2 x x g(x) f(x) e 1 , x 1,1 2 = + + , ( )x g (x) f (x) e x 0 , x 1,1 = + < ) x x f (x) 0 , e x x e 0 < > < , g( ) ''1 1'' ( 1,1) 2 , g(0) f(0) 1 0 1 0= + + = x 0= , g(x) 0= 19 z z 1 1 z x = =c ) x 1 z AM= = , 2 0 x AM 2 = , x 0= A M x 2= B M , [ ]x 0,2 ) ( ) ( ) 2 22 2 2 2 2 1 z x 1 z 1 z x 1 z z zz x 1 z z z x z z 2 x = = + = + = + = ) ( )2 2 1 z zz z z z z z z z z z+ = + = + = + = + 16. 16 ) )2 2 2 2 x x 2 , x 0, 2 1 z 1 z x z z x 2 x x x 2 , x 2 ,2 + + + + = + + = + = + , ) ) ( 2 2 2x 1 , x 0, 2x x 2 , x 0, 2 f(x) , f (x) x x 2 , x 2 ,2 2x 1 , x 2 ,2 + + + = = + + ( )1 9 f(0) 2 , f , f 2 2 , f(2) 4 2 4 = = = = f f 2 , f 4= = 20 f : r r x 2 0 dt f(x) 2 1 3f (t) = + ) ( )2 2 f(0) 0 , f (x) 0 , f 1 3f (x) = = > + 1 r (1) f x 0 f(x) f(0) f(x) 0> > > 1 , f x 0 f(x) f(0) f(x) 0< < < 1 ( ) 22 2 f (x) 6f(x)f (x) 1 3f (x) = + , ( ) 22 1 3f (x) ,f (x) 0+ > f f x 0= f(0) 0= ) (1): ( )2 3 3 f (x) 3f (x)f (x) 2 f(x) f (x) (2x) f (x) f(x) 2x c + = + = + = + x 0= 3 f (0) f (0) 0 c c 0+ = + = , 3 f (x) f(x) 2x , x+ = r ) ) ( )f1 r ''1 1'' 3 f(x) y y y 2x= + = ( )31 x y y , y 2 = + r ( )1 1 31 f :f( ) f (x) x x 2 = +r r ) 1 1 0 0 E f(x) dx f(x)dx= = ( ) ( )1 1 21 u f(x) f (u) x , dx f (u) du dx 3u 1 du 2 = = = = + , u f(0) 0 , u f(1) 1= = = = ) (.Horner) 3 3 f (1) f(1) 2 f (1) f(1) 2 0 f(1) 1+ = + = = , ( ) ( ) 14 21 1 2 3 0 0 0 1 1 1 3u u 5 E u 3u 1 du 3u u du 2 2 2 4 2 8 = + = + = + = = K 21 f(x) , 0 x 1 f (0) f (1) 0 , 0 f(0) f(1) , g(x) x 0 , x 0 < = = = < = = 17. 17 ) x 0 x 0 x 0 f(x) f(x) f (0) g(0) 0 , lim g(x) lim lim f (0) 0 x x 0+ + + = = = = = ) ( ) 2 x 1 x 1 x 1 f(x) f(1) f (x)x f(x) g(x) g(1) f(x) f(1)xxx 0,1 g (x) , g (1) lim lim lim x x 1 x 1 x(x 1) = = = = = _ _ x 1 x 1 f(x) f(1)x f(1) f(1) 1 f(x) f (1) f(1)(x 1) lim lim f (1) f(1) f(1) 0 x(x 1) x x 1 x(x 1) + = = = = < ) g 1 g(x) g(1) , g(x) g(1) 0 x 1 0 < , x 1 g(x) g(1) g (1) lim 0 x 1 = , . g (1) 0 ) 0 g(0) 0 f(1) g(1)= < = . g(0) . g [0 , 1] [0 , 1] ) ( )0x 0,1 g . Fermat ( )0 0 0 0g x 0 f (x )x f(x ) 0 (1) = = 0 0 0 0 0 0 0 0 0 : y f(x ) f (x )(x x ) , O () f(x ) f (x )( x ) f (x )x f (x ) 0 = = = (1) 22 [ ]f :, r , f (x) 0 ( )x, 1z e if()= + , 2z f () ie= + 2 2 4w w , 3 3w = r ) w 4 3 3w 4w 0 + = . 4 3 g(x) 3x 4x= + , x r 2 g (x) 12x (x 1) = 0 x 0 x 1= = = g g(1) 1= x x lim f(x) lim f(x) + + = = + g [ )g(A) 1,= + , g(x) 0= 1 0 1 > > ) ( ) 2 2 2 2 1 0 , z 1 e f () 1 e f () 1 = + = + = . 2 2 2 0 e 1 f () 0 1 0 f(0) 0, z 1= = = ) ( ) 1 2z z e f() f()e e f()f( i+ = + + , 1 2Im(z z ) 0 f()f() e 0+ = = < , f . Bolzano 0 0x (,) f(x ) 0 = f (x) 0 f 1-1 0x . ) () 1 2 f() f() u z z I e e = = x f(x) h(x) e = . Rolle [ , ], ( ) , h () 0 f () f () = =LL 23 x f(t) t , t 0= > x 1 f (t) xt = ) ... 1 (3,4) , x 1 x x 1 1 f(4) f(3) f ( ) x 4 3 4 3 = = 18. 18 ) x 1 () x x x x x 1 x 1 1 1 1 2 2 2 4 3 6 5 x x x 0 0 0 = = = = x 1 0 x 1 = = ) ( ) 1 2 1x 1 (x 0) x x x x x 1 x 1 1 1 2 2 4 5 3 6 x 0 1 x 1 0 x 1 < > + > + > > < > L ( ) 1x x x x1 x x x x 0 0 4 5 3 6 3 4 2 5 4 5 3 6 dx 0 0 ln 4 ln5 ln3 ln 6 ln 4 ln5 ln3 ln 6 + > + > + > + L ) g ( ) ( )g g 1 2 g . () 1 2(2 x 3 4 x 5)(... 1 2 1 2g(3) g(4) g(2) g(5) g(3) g(2) g(5) g(4) g (x ) g (x ) x x < < < < < + > + > > < , g ( )1 , ( )2 , g 24 ( )f : 0, , f(1) 1 , f(x) 0 ,x 0 xf (x ) 2f(x) 0,+ = > + ) f(x) 0 , , f(1) 1 0= > , ( )f(x) 0 , x 0,> + . ( ) f (x) 2 xf (x) 2f(x) lnf (x) 2ln x 0 f (x) x < < + < , ( )2 g(x) lnf(x) 2ln x ln x f(x)= + = ( )g (x) 0 x 0, < + , ( )g 2 ( )0,+ , ( ) ( ) g 2 2 1 0 x 1 g(x) g(1) ln x f(x) 0 f(x) x < < > > >L 2 ( )g 2 1 x 1 g(x) g(1) f(x) x > < = + = + , x 0 (yy )= x 0+ ( )3 x 3x f(x) 1 0 , x 1, f(x)x x lim 0 1 x lim 0 x + + < < + = = , ( )2 x 2x 1 0 f(x) , x 1, x lim f(x) 0 1 lim 0 x + + < < + = = y 0 (xx )= x + ) ... f 1 ,2 2 1 f(2) f 5 5 1 152 2f () 5 0 f () f (2) f 12 2 2 42 2 + < < < < 19. 19 ) ( ) 1 0 f(2) 4 1 1 15 f(2) f 4 1 1 2 4 4 f 4 f 4 2 2 + < < < = > < ) 1 1 f(x)dx , 0 1 E() f(x)dx , 1 < = > , ( ) 11 1 0 1 2 2 1 1 1 1 x 0,1 ,f(x) f(x)dx dx E() 1 x x x < < > > > = 0 1 lim 1 + = + 0 lim E()+ = + ( ) 1 2 2 1 1 1 1 1 1 1 x 1, f(x) f(x)dx dx E() 1 1 0 x x x > + < < < = < > , 25 ( ) x f (x) x 2 e = , x r , : y 2x 1= + 2 2 4 5 z ,z z z = c ) x f(x) lim 2 (1) x+ = ( )x lim f(x) 2x 1 (2) + = ( ) [ ] ( ) ( ) ( ) x x x xxxt t t t 0 00 0 0 0 f (t)dt t 2 e dt f (t) t 2 e dt f (x) f (0) t 2 e e dt = = = + f (0) c x f (x) e (1 x) 1 c = = +L ( ) x x t 0 0 f (t)dt e (1 t) c 1 dt = + x x x t t 0 0 0 f(x) f(0) e dt te dt (c 1)dt = + L f (0) k x f(x) xe (c 1)x k = = + + , (1) x x x f(x) k lim lim e c 1 2 c 1 c 3 x x + + = + + = = , ( ) ( ) (2) x x x lim f(x) 2x lim xe 2x k 2x 1 k + + = + + = x f(x) xe 2x 1 , x = + r ) x x 2e x 1 f (x) e + = , x g(x) 2e x 1,x= + r x g (x) 2e 1 0 x ln 2 = = = g(x) g( ln 2) 0 > , x e 0> f (x) 0 > ( )f 1 1-1 . x x lim f(x) , lim f(x) + = = + 1 f D f(A) ( , ) = = + = r ) f "1 1"4 4 f(x 4x 5) 1 f(0) x 4x 5 0 + = = + = . 4 3 h(x) x 4x 5 ,x ,h (x) 4x 4 0 x 1= + = = =r x 1= h h(1) 2= x x lim h(x) lim h(x) + = = + [ )h(A) 2,= + , 0 . 20. 20 2 4 2 4 5 z z 4z 5 0 z z = + = , z 4 x 4x 5 0 + = , ) 1 1 e 1 1 E f (x) dx + = , 1 u f (x) x f(u) = = dx f (u)du= , f "1 1" 1 f "1 1" 1 x 1,u f ( 1) f(u) 1 f(0) u 0 1 1 1 x 1 ,u f (1 ) f(u) 1 f (1) u 1 e e e = = = = = = + = + = + = = [ ] 1 1 1u 0 1 0 0 0 0 3 E u f (u)du uf (u)du uf(u) f(u)du e > = = = = = L 26 A f ( ) x 2 1 f (t) 0, f(x) e dt t + = ) 1 1 1 1 x x x x 2 2 f (x) 1 f (x) f (x) f(x) 0 e f (x) e f(x) 0 e f(x) 0 e f(x) c x x = + = + = = = 1 1 x x 2 1 f(1) e c 1,f(x) e ,f (x) e 0 x = = = = < ( )f 2 , 1 1 x x 3 4 2 1 f (x) e e 0 x x = + > f ) ( )0x 1,2 x 1 12 x F(x) f (t)dt1 f(x)dx e F(2) F (x) F (x) F(2) 0 = = = = , Bolzano G(x) F(x) F(2)x= , . . Rolle. G(1) F(2) , G(2) F(2)= = . Rolle ( )0x 1,2 0 12 x 0 1 G (x ) 0 f(x)dx e = = L . ( )G (x) f(x) F(2),G (x) f (x) 0,G = = < 2 0x ( ) ... F [1 , 2] ) x 1 f (t)dt ex e F(x) ex e F(x) ex e 0= = + = ( )h(x) F(x) ex e,x 0,= + + h(1) F(1) e e 0= + = , h (x) F (x) e f(x) e f(x) f(1) = = = , ( )f x 1 f(x) f (1) h (x) 0> < < 2 , ( )f 0 x 1 f(x) f(1) h (x) 0< < > > 2 , x 1= ( ) : y F(1) F (1)(x 1) y ex e = = , F (x) f(x) , F (x) f (x) 0 = = < ) F . F(x) ex e< " "= x 1= ) 3 x x f(t) G(x) dt t 1 = , x 1 x 1+ > 3 x x< 21. 21 ( ) 3f (t 1 0) 3 3 f(x) f(t) f(x ) x t x f(x) f(t) f(x ) t 1 t 1 t 1 > 2 3 3 3 3x x x x x x f (x) f(t) f(x ) dt dt dt t 1 t 1 t 1 , 3 3 x x 3 x x 1 1 f(x) dt G(x) f(x ) dt t 1 t 1 ( ) ( )2 3 2 f(x)ln x x 1 G(x) f(x )ln x x 1 + + + + ( ) ( )2 3 2 x 1 x 1 lim f(x)ln x x 1 lim f(x )ln x x 1 eln3+ + + + = + + = , x 1 lim G(x) eln3+ = 27 f , g : r r x x 2 1 1 f (t)dt g(t)dt x 2x 1 , x= + + r ) f(x) g(x) 2x 2= + , y x= + ( )x x f(x) lim 1 lim f(x) x 0 x+ + = = , x x g(x) f(x) 2 lim lim 2 1 2 0 1 x x x+ + = + = + = ( ) ( )x x lim g(x) x lim f(x) x 2 0 2 2 + + + = + = + = , y x 2= + g + ) 1 2 1 2x 1 x , g(x ) g(x ) 0< < = = , 1 1 1 1f(x ) g(x ) 2x 2 2(x 1) 0= + = < , 2 2f(x ) 2(x 1) 0= = >L , . Bolzano 0 1 2 0x (x ,x ) , f(x ) 0 = (1) ) 2 2 1 2 0 0 0 1 E f(x) g(x) dx 2 x 1 dx 2 (1 x)dx (x 1)dx 2 = = = + = = L ) 0 2 2 0 2 2 0 2 2 f(x) 2x x x f (x) f (x)(x x) f (x) 2x x x f (x)(x x) f (x) 2x (x x ) x x + = = + = + ( ) ( )2 2 0 2f(x)(x x) x (x x )x = + 2 2 0 2H(x) f(x)(x x) x (x x )x= + + [ ]0 2x x ,x ( )1 2 0 0 0 2 0 2 0H(x ) x (x x )x x x= + + = , 2 2 2 0 2 2 0 2H(x ) x (x x )x x x= + + = , 0 2H(x ) H(x )= , .Rolle ( )0 2 x ,x 0 2 2 f() 2 x x H () 0 f () x + = = L 28 x1 xf (x) f(x) xf (x) ,x 0 (1) g(x) e x x + + = > = ) ( ) (0 x) 2 2 1 xf(x) x f (x) 1 xf (x) x x f (x) xf(x) 1 (x 1)f (x) f(x) x < + = + + = + = ( ) ( )(x 1)f(x) ln x (x 1)f(x) ln x c = = + , (1) f(1) 1= c 0= ln x f(x) ,0 x 1 x 1 = < , x 1= f 1 0 0 x 1 x 1 DLH ln x f(1) limf(x) lim 1 x 1 = = = , ln x , 0 x 1 f(x) x 1 1 , x 1 < = = ) ( ) ( ) ( )2 1 1 ln x xf (x) ,x 0,1 1, x 1 = + ( ) 2 1 1 x h(x) 1 ln x , x 0, , h (x) x x = + = 22. 22 0 x 1,h(x) h(1) 0< < = f (x) 0 < f 1 ( )f 2 ( )0,+ "1 1" x x DLH xx 0 x 0 ln x ln x 1 lim f(x) lim , lim f(x) lim lim 0 x 1 x 1 x+ + + + + = = + = = = ( ) 1 f f(A) 0, D = + = ) 1 2x x< ( ) ( )1 1f g(x 1 f g(x= = ( ) ( ) f "1 1" 1 1 1 2f g(x f(1) f g(x g(x ) g(x ) 1 = = = = (1) ( )x x x ex 1 0 x e 0 x e 0 + = + = = [ ]x 1 2F(x)x e , x x ,x = 1 1 1 1 1 x (1) x 1 1 1 1 1 x x x ex 1 g(x ) 11 F(x )x e x 0 e e e = = = = = , 2F(x ) 0= =L F .Rolle ( )1 2 x ,x F () 0 e 1 0 = + =L ) x 0 x 0 u 1 u 1 x 1, lim(1 x) 1 , lim f (1 x) lim f(u) f(1) 1+ + + = + > + = + = = = , 5 5 x 0 lim f (1 x) 1 1 0+ + = = > ( ) 3x 2 x 0 lim e 1 x 0+ = , x 0+ ( ) 3x 2 e 1 0 , x 0 > > ( ) 3x 2x 0 1 lim e 1 x + = + , ( ) 5 3x 2x 0 f (1 x) lim e 1 x + + = + 29 2 1 4 x 1 f(x) ln x x , g(x) ln x 2 , h(x) x 1 ln xx = + = + = + ) ( ) ( ) 2 x 11 1 1 f (x) 0 , x 1, x 2 x 2x x 2x x = = = < + L , ( ) [ )f 1, , x 1 f (x) f(1) f (x) 0+ > < + + + L ( ) [ )g 1, , x 1 g(x) g(1) g(x) 0+ > > >1 ) (ln x 0) x 1 x 1 1 x x ln x x 1 ln x ln x x 0 f(x) 0 ln x x x x > < < < + < < ) (ln x 0) x 1 x 1 2x 2 4 2x 2 (x 1)ln x ln x 2 ln x 0 g(x) ln x 2 x 1 x 1 > + < < + < < < + + ) ) 2 22 2 2 3 3 3 2 2 2 x 1 x 1 E h(x) dx dx dx ln x ln x = = = , ) 2 2 2x 1 x x : 2x x 1 ln x < < + 232 222 2 33 3 322 2 2 7 x 49 2xdx x , (x 1)dx x 4 3 24 = = + = + = , 7 49 E 4 24 < < ) x 3 2 F(x) h(t)dt= 2F(x) 4x 5 0 + = , 3 (x) 2F(x) 4x 5 ,x ,2 2 = + , 3 1 0 , (2) 2F(2) 3 0 2 = < = > , () 7 3 F(2) E 4 2 = > > 23. 23 .Bolzano 3 ,2 2 () 0 2F() 4 5= = (1) (1) 2 2 2 2 4 5 2F() h(x) 1 F (x) 1 F (x) F (x) 2 3 2 3 2 3 2 3 = = + = = 3 F() F 2 F (x) 3 2 = , ... F 3 , 2 3 F() F 2 22 F () h() 1 3 2 3 2 = = L 30 f ( )0,+ 2 x 2 2 2 x 4 t 3x tf dt 2x ln x 4x 1 , (1) x 0 x x + = + > ) t u t xu , dt xdu x = = = 2 x x 2 x 1 t tf dt x uf (u)du x = , (1) x 2 2 1 3x uf(u)du 2x ln x 4x 1+ = + 6x 4xf(x) 4x ln x 2x 4+ = + + 1 f(x) ln x 1 x = + ) 2 x 1 f (x) , x 0 x = > x 1= f f(1) 0= ) 3 2 x f (x) ,x 0 x = > x 2= f 1 f(2) ln 2 2 = + , : 1 y f(2) f (2)(x 2) y x 1 ln 2 4 = = + ) 2 , f f ( ) e2 2e 2 2 1 1 x e 7 E x ln 2 ln x dx x ln 2 x ln x x ln x e 1 ln 2 4 x 8 8 2 = + = + + = + + 31 f f (x) e f (x) x 1 (1) , x+ = + r 24. 24 ) (1) x 0= f (0) e f(0) 1 0 (2)+ = , x x h(x) e x 1 , h (x) e 1 0 , h( )= + = + > 1 r , "1 1" (2) ( ) h"1 1" h f(0) 0 h(0) f(0) 0 = = = , ( ) ( ) ( )x x h lim h(x), lim h(x) , + = = + =r r ( )f (x) 1 f (x) 0 , f e 1 = > + 1 , "1 1" ( ) ( ) h"1 1" 1 1 h f(x) x h h (x) f(x) h (x) = = = x 1 h(x) e x 1 f (x) = + = ) ( ) f (x) 2f (x) 1 f (x) e f (x) 0 e 1 = < + f , g(x) xf (x) f(x),x 0,g (x) xf (x) 0 = = < ( )g x 0 g(x) g(0) xf (x) f(x) 0 xf (x) f(x) > < < < 2 , x 1 k(x) f(x) ,x 0, k (x) f (x) 0 2 2 = = < ( )f f (x) f (x) f (x) 1 1 1 x 0 f(x) f(0) f(x) 0 e 1 e 1 2 f (x) e 1 2 2 > > > > + > < < + 1 ( )k x x x 0 k(x) k(0) f(x) 0 f(x) 2 2 > < < < 2 ) x 0> f(x) 0> , 1 0 E f(x)dx= , ) f(x) xf (x)> [ ] 1 1 1 1 0 0 0 0 f(x)dx xf (x)dx E xf(x) f(x)dx E f(1) E 2E f(1)> > > > ) ( ) x 0 F x f (t)dt= F (x) f(x) = , 2x 7x x 3x f(t)dt f (t)dt 0 F(2x) F(7x) F(x) F(3x)+ = + = + ( )x 0 ,F (x) f(x) 0, F> = > 1 ( )F 0 x 2x F(x) F(2x) F(x) F(3x) F(2x) F(7x) 0 3x 7x F(3x) F(7x) < < < + < + < < < 1 ( )x 0 ,F (x) f (x) 0, F< = < 2 ( )F 2x x 0 F(2x) F(x) F(2x) F(7x) F(x) F(3x) 7x 3x 0 F(7x) F(3x) < < > + > + < < > 2 x 0= . 32 2 2 f(x) 1 x 2x ln x= + , 2x 2 1 1 2t g(t) g(x) dt (1) , x 0 t(1 t ) = > + ) f (x) 4x ln x = x 1= f f(1) 2= , f(x) 2 , x 0 > ) ( ) ( ) 2 2 DLHx 0 x 0 x 0 x 0 2 2 ln xln x x lim x ln x lim lim lim 0 1 21 x x + + + + = = = = , x 0 lim f(x) 1+ = , ( )( )2 x x lim f(x) lim 1 x 1 2ln x + + = + = , ( ] ( ) ( ] ( )1 1 1A 0,1 , f A 1,2 , 0 f A= = [ ) ( ) ( ] ( )2 2 2A 1, , f A ,2 , 0 f A= + = 2A f() 0= f ( )2 . 25. 25 ) (1) 2 2 2 2 2 1 2x g(x) 1 g (x) g (x)x(1 x ) 2x g(x) 1 g (x)(1 x ) 2xg(x) x(1 x ) x = + + = + + = + ( ) ( ) ( )2 2 2 2 g (x)(1 x ) (1 x ) g(x) ln x g(x)(x 1) ln x g(x)(x 1) ln x c , g(1) 0 + + + = + = + = + = c 0= 2 ln x g(x) 1 x = + ) 2 2 f(x) g (x) x(1 x ) = = + L , g f ( ]1x A 0,1 = ) f(x) 0> , ( )f 1 x f(1) f (x) f() f(x) 0< < > > > 2 , ( )f x f(x) f() f(x) 0> < < 2 , g x= 2 ln g() 1 = + , 2 2 f() 0 1 2 ln = + = 2 2 ln 1 g(x) g() 2 ln 2 = = 33 f ( ) ( ) 0 x 2 4 tf x 2t dt ln x 1 x , x 1 = + > ) x u du u x 2t t , dt , 2 2 = = = ( ) 0 x x x x 0 0 0 2 x u du 4 tf x 2t dt 4 f(u) x f(u)du uf(u)du 2 2 = = = + L x 0 1 f (u)du 1 1 x = + , 2 1 f(x) 1 x = + ) ( ) ( ) xx 0 e 1 F(x) f (t)dt x x F (x) f (x) 0 , F ,"1 1" ln(x 1) f(t)dt 0 F(e 1) F ln(x 1) e 1 ln(x 1), = = > + = = + = + 1 x e 1 1 > , ( ) 1 1 ln x 1 1 x 1 e x 1 e + > + > > x k(x) e ln(x 1) 1,x 1= + > , ( ) ( )x x 2 x 1 1 1 1 k (x) e ,x 1 , , k (x) e 0 , k ,k (A) k 1 , lim k (x) x 1 e ex 1 + = + + = + > = + + + 1 ( ) 1 1 e 1 k (A) e , , k (x) 0 k 1 , e + = + > = + + 1 k(0) 0= x 0= ) ( ) ( ) 2 3 2 2 1 x g(x) x x , x 1 x 1x 1 = + = > ++ , g(x) 0 x 0= = 26. 26 12 21 1 0 0 0 x 1 x 1 E dx x 1 dx x ln(x 1) ln 2 x 1 x 1 2 2 = = + = + + = = + + + L ) ( ) ( )2x x 3x 2x x x x 2x e e 1 h(x) e e f(e )e e 1 e + = + = + , x e u = , x x x x lim e 0 , lim e + = = + ( )2 1 2 2x u 0 u 0 u u 1u (u 1)u lim h(x) lim lim 1 1 u (1 u )u+ + + + + = = = + + , : y 1= + , ( )2 1 2 2 2x u u u u u 1u (u 1)u u 1 u lim h(x) lim lim lim 0 0 0 1 u (1 u )u 1 u u + + + + + + = = = = = + + + , y 0= . ( )u 0 uu 1 1 u 1 u u u u u u > = u u lim 0 u+ = 34 f [ )0,+ f (x) 0 x 0> > , ( )x 2 2 1 t 1 f(t)1 f(x) dt,x 0 ex x = > x 1 F(x) f(t)dt,x 0= ) ( ) x 2 1 1 x f(x) t 1 f(t)dt e = , ( ) ( )2 2 f (x) 1 3 1 1 x f (x) 3x 1 f(x) 0 ln f(x) 3ln x lnf (x) 3ln x c f(x) x x x x + = = = = + 1 f(1) c 0 e = = 1 1 3ln xx x 3 1 1 ln f(x) 3ln x f(x) e e e , x 0 x x = = = > . f 0 ( ) 1 1 u 3 2x x 3 u u 3 u u uux 0 x 0 x 0 x 0 x 0 x 0 e u 3u 6u f(0) lim f(x) lim lim u e lim lim lim lim 6e 0 x e e e+ + + + + + = = = = = = = = = ) 2 1 3x f (x) f(x) ,x 0,f(x) 0 x = > > f 1 x 3 = 3 1 27 f 3 e = x f(0) 0 , lim f(x) 0 + = = 3 27 f(A) 0, e = ( ) ( )3 3 27 e f x ln x 1 27 0 f x ln x 1 e + = + = x 1 ln x 0,x ln x 1 x ln x 1 0> > + > + > 3 27 e f 1 x 3 = 1 x ln x 1 3x 3ln x 4 0,x 1 3 + = + = > g(x) 3x 3ln x 4= + , g(1) 1 0 , g(e) 3e 1 0= < = > . Bolzano 0x (1,e) 0g(x ) 0= ( ) 3 g (x) 3 0 , g x = + > 1 ) 2 1 1 u , du dt t t = = 1 11 1x x xu u ut x 3 11 1 1 1 F(x) e dt ue du u(e ) du ue t = = = = + 1 1 x u x 1 1 2 e du 1 e , x 0 x e + = = + > L 27. 27 ) ( ) 1 u1 x ux x x u 0 1 2 2 2 lim F(x) lim 1 e lim 1 u e 1 x e e e = + + = + = = ( ) 3 2 2 3 2 e F (x) f(x) 0 , F , 1 x 2 F(x) F(2) F(x) 1 0 e e2 e 2 e = > < < < = < < 1 , 1 1 12 2 2 x x x 1 1 1 2 1 1 E F(x) 1 dx 1 e 1 dx e e 1 dx e x x = = + = + = ( ) ( ) ( ) 1 1 1 1 1 12 2 2 x x x x x x 2 1 1 1 1 1 x e xe 1 dx x e xe 1 dx x e x e 1 dx x x = + = + = + = ( ) 2 21 12 x x 1 1 e 11 2 xe x dx xe x 1 e ee = = = + = 35 f r 2f(2x) f(x) 2x = (1) ( ) 2 2 1 F(x) x f xt dt x 3 (2), x= + r ) f ... ( ) ( )1 2 0,1 1,2 1 2 f(1) f(0) f(2) f(1) f ( ) 2f ( ) 2 2f(2) f (0) f(1) 0 1 0 2 1 + = + = = x 0 x 1 (1) f(0) 0 , (1) 2f(2) f(1) 0 = = = = ) u tx , du xdt= = 2 2x 2x x 1 x 0 0 x f (xt)dt f(u)du f(u)du f(u)du= = (2) 2x x 2 0 0 F(x) f(u)du f(u)du x 3= + (1) F (x) 2f(2x) f(x) 2x 0 = = F F(0) 3 F(x) c c 3 , F(x) 3 = = = = ) 2 2 2x 2x x 1 12 2 4x 2x x 2 2 f(u)du 1 1 F(x) 3 f(u)du x 3 3 f(u)du x f(u)du 2 4 = = = = = + = = = = 4 2 4 1 1 2 I f(x)dx f(x)dx f(x)dx 1 4 5= = + = + = ) ( ) x 0 x x x x x x 0 x 0 0 0 2 f(2t)dt f(t)dt e 2f(2t)dt f(t)dt e 0 2f (2t) f(t) dt e 0+ = = = x xx 2 x 2 x 00 2tdt e 0 t e 0 x e 0 = = = , [ ]2 x 1 g(x) x e ,x 1,0 , g(0) 1 0 , g( 1) 1 0 e = = < = > . Bolzano ( )1,0 g(a) 0= [ ] ( )x g (x) 2x e 0 x 1,0 ,g = < 2 , . 28. 28 36 f [ )0,+ , x x 0 1 lim f(t)dt 0 x+ = (1) ) x 0 F(x) f(t)dt , F (x) f(x) ,F(0) 0= = = x 2x 0 x 1 1 F(x) F(0) F(2x) F(x) f(t)dt f(x) f (t)dt F (x) x x x 0 2x x < < < < , ... F ( )1 2 0,x (x,2x) 1 2 1 2 F(x) F(0) F(2x) F(x) F(x) F(2x) F(x) F ( ) F ( ) f( ) f( ) x 0 2x x x x = = = = (2) ( )f x 2x(2) 1 2 1 2 0 x 1 1 0 x 2x f( ) f(x) f ( ) f(t)dt f (x) f(t)dt x x < < < < < < < < 1 ) ( ) )x f , f(A) f(0), lim f(x) + = 1 (1) x x x0 1 F(x) lim f (t)dt lim 0 x x+ + = = , 2x x x xx 1 F(2x) F(x) F(2x) F(x) lim f(t)dt lim lim 2 2 0 0 0 x x 2x x+ + + = = = = , ) x lim f(x) 0 + = [ )f(A) f(0),0= f(x) 0< ) f(x) 0< 4 1 E f(x)dx= 4 1 f(1) 2f(2) f(x)dx+ < , ) 1 2 0 1 x 1: f(t)dt f(1) f(t)dt (3)= < < 2 4 4 0 2 2 1 1 x 2: f(t)dt f(2) f(t)dt 2f (2) f(t)dt 2 2 = < < < 2 4 4 1 2 1 f(1) 2f(2) f(x)dx f(x)dx f(x)dx+ < + = ) x 0 f(x) f(t)dt 0 = , x 0 g(x) f(x) f(t)dt= 1 0 g(0) f (0) 0 , g(1) f(1) f(t)dt 0= < = > (3) , . Bolzano ( )0,1 0 g() 0 f() f(t)dt F()= = = (4) ... F ( ) 0, (4) F() F(0) f() F () f() f() f() 0 = = = , ( )f() f() f () 0< < , f () 0< < 37 f r , , f(0) 0= , 0x 0= : y x 2= + f (0) 1 = , ( )2 x 1 f (x) 4xf (x) 2f(x) 0,x + + + = r 2 g(x) x f (x) ,x= r 29. 29 ) (1) 2 2 g (x) 2xf(x) x f (x) , g (x) 2f(x) 4xf (x) x f (x) f (x) = + = + + = , g (x) f (x) c = + g(x) f(x) cx k= + + , f (0) 1 = f(0) 0= c 1,k 0= = 2 2 x g(x) f(x) x x f(x) f(x) x f(x) x 1 = + = + = + ) ( ) ( ) ( ) 22 2 32 2 2x 3 x1 x f (x) , f (x) x 1 x 1 = = + + 3 f( 3) 4 = , 3 f( 3) 4 = ,f(0) 0= 1 f(1) 2 = , 1 f( 1) 2 = ) ( ) ( ) ( ) 2 23 22 2 x x 3x g(x) , g (x) 0, g x 1 x 1 + = = > + + 1 ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) 2 3 2x 2 x x 2 x 333e 1 1 ln x 1 e 1 1ln x 1 e 1 1 ln x 1 e 1 1 ln x 1 + + = + + + + = + + ( ) ( ) ( ) ( ) ( ) 3x 3 x x 2 2x e 1 ln x 1 g(e 1) g ln(x 1 e 1 ln(x 1) ln x 1 1e 1 1 + = = + = + + + + , ( ) x x x 2 1 1 h(x) e 1 ln(x 1), x 1 , h (x) e , h (x) e 0 , h ( ) x 1 x 1 = + > = = + > + + 1 ( ) [ )x 0 h (x) h (0) h (x) 0 ,h 0, > > > + 1 ( ) ( ]1 x 0 h (x) h (0) h (x) 0 ,h 1,0 < < < < 2 , h(0) 0= x 0= ) 2 x 3x 3 0 2x g(t)dt 2x 1 + = , 2 x 3x 3 0 G(x) 2x g(t)dt 2x 1 + = + 31 1 1 2 2 0 0 0 t t G(0) 1 0 ,G(1) 2 g(t)dt 1 2 dt 1 2 t dt 1 t 1 t 1 = > = = = = + + ( ) 12 2 0 t 1 ln t 1 1 ln 2 0 2 2 = + = < , . Bolzano ( )0,1 G() 0= 2 3 3 0 2 1 g(t)dt 2 + = 38 f f(x) 0 , x> r [ ] 2 x 0 0 f(x)dx 1 , g(x) f (t) f(2 t) dt , x= = + r ) f ( )f 1 , ( )g (x) f(x) f(2 x) 0 ,g ,g(0) 0,x 0 g(x) 0 ,x 0 g(x) 0 = + > = > > < > > > 1 ( )f g (x) 0 f (x) f (2 x) x 2 x x 1 < < < < 1 x 1= g g(1) [ ] 1 2 1 2 1 0 0 2 0 2 g(1) f(t) f(2 t) dt f(t)dt f(t)dt f(2 t)dt f (2 t)dt= + = + + + (1) 2 0 2u 2 t du dt0 2 0 f(2 t)dt f (u)( du) f(u)du 1 = = = = = (2), 1 1 1u 2 t du dt2 0 0 f(2 t)dt f (u)( du) f(u)du = = = = (1) 2 1 1 2 2 1 0 0 1 0 g(1) 1 f(t)dt 1 f(u)du 2 f(u)du f(t)dt 2 f(t)dt 2 1 1 = + = + = = = ) 2 2 2 2 (2) 0 0 0 0 E g (x)dx g (x)dx f(x)dx f(2 x)dx 1 1 2 = = = + = + = ) x 0 F(x) f(t)dt , F (x) f(x)= = ( )xF (x) F(x) 1 xF(x) x 0 + = = G(x) xF(x) x= 2 0 G(0) 0 , G(2) 2F(2) 2 2 f(t)dt 2 2 1 2 0= = = = = , . Rolle ( )0,2 0 G () 0 F() F () 1 0 f() f(x)dx 1 = + = + = L 39 ( ) x f(x) , x 0, x = ) 2 xx x f (x) ,x 0, x 2 = h(x) xx x ,x 0, 2 = h (x) xx 0,x 0, 2 = < ( ) h 0, , 0 x h(0) h(x) 0 xx x 2 2 < < > > 2 , f (x) 0,x 0, 2 < ( ) f 0, 2 2 x 2 2 x f(x) fx x 2 2 x < > > > 2 2 4 4 x x E dx dx x x = = , x 2 x > 2 2 4 4 x 2 2 1 1 dx dx x 2 4 2 2 > > = > ) ( ) ( ) x 2 4 4 4x 3 f(t)dt 1 x x = , ( ) ( ) x 2 4 g(x) 4 4x 3 f (t)dt 1 x x= 2 4 3 3 g 0 , g(1) 4 f(t)dt 4E 2 0 4 4 4 = < = = > > . Bolzano 3 ,1 4 g() 0= L 2 4 4 f(t)dt 1 4 3 = ) ( ) ( ) 0ux 2 2 2 2 2 2 2 1 dux dx 3 3 3 3 2 x1 1x 1 E dx dx dx dx du xf(x)x x 1 x 1 u = = = = = = = = 31. 31 ( ) ( ) 1 1 1 2 2 2 0 0 0 1 1 1 A B 2 2du du du u 1 1 u u 1 1 u u 1 1 u = = = = + + + 1 2 0 1 1 1 du 2 u 1 u 1 = = + 1 2 0 1 u 1 ln3 ln 2 u 1 2 = = = + L ) 2 2 2 2 2 2 4 4 2 2 22 2 2 2 2x 0 x 0 x 0 x 0 4 1 x x x x 1 1 x x 1 3x xlim 1 lim lim lim x xx f (x) x x 1 3x x x = = = = = 0 0 22 2 2 2 20 0 4 3 2 2x 0 DHL x 0 DHL x 0 x 0 x 0 x x xx x x x 1 2 x 1 x 1 lim lim lim lim lim x 2x 6x 6x 3 x 3 = = = = = 40 f [ )0,+ 1 2 x 2 f(1) , x f (x) f(x) e 0 , x 0 e = + = > z z 0 ) 1 1 1 1x 2 x x x 2 2 2 2 1 e 1 1 x f (x) f(x) e 0 f (x) f (x) 0 e f (x) e f(x) x x x x + = + + = + = 1 1 1 x x x 1 1 2 1 e f(x) e f(x) c , f(1) c 1 , f(x) 1 e , x 0 x x e x = = + = = = + > f 0 1 y 1 x yx yy yx 0 x 0 y DLH 1 1 y f(0) lim f (x) lim 1 e lim lim e 0 x e+ + = = = + = = = ) ( ) 1 1 1 1 1 11 1 1 1 x x x x x x 1 1 E 1 e dx e e dx x e x e dx xe dx x x = + = + = + = = 11 1 x 1 xee e = = ) ( ) 1 x 3 e f (x) 0 , x 0, x = > + ( ) [ )f 0, + 1 1 y 1 x y 0x x x y y 0 1 lim f(x) lim 1 e lim (1 y)e 1 e 1 x = + + = + = = = y 1= ( ) [ ) ( ) ) [ )x f 0, , f A f (0), lim f(x) 0,1 + = + = = 1 f(x) 1< ) zz 1 1 1 1 1 1 1 1 , 1 1 z z z z z + + = + + + , z 1 1 z z1 1 1 1 1 1 1 e z ln 1 1 ln 1 1 e 1 e 1 z z z z z z + < + < + < + < + < 32. 32 ( )f z 1< ) z 0> 41 f f(0) 1= , [ )0,+ , , , ( )f 1 22 t u tu tx t tu 0 t 0 t tt t DLH e e t lim lim lim 21 u 2e f(t)dt , x 0 tF(x) x 1 , x 0 1 1 lim lim e 0 2e 2 + = = = > = = = = = = ) x 0= , ( )f x 0 , 0 t x f(0) f(t) f(x) 1 f(t) f(x)> 1 " "= x x x x xx 0 0 0 0 0 0 1 dt f(t)dt f(x)dt x f(t)dt xf(x) 1 f (t)dt f(x) x > < < < < < < . 1 F(x) f (x)< < , 1 F(x) f (x) , x 0 ) [ ) x 0 x 2 0 xf(x) f(t)dt , x 0 F (x) , G(x) f(t)dt ,x 0, , G (x) f(x) f(0) 1 0x 1 f (0) , x 0 2 > = = + = > = > = 0 0 2 x 0 x 0 x 0 x 0 DLH F(x) F(0) G(x) x G (x) 1 f(x) f(0) 1 F (0) lim lim lim lim f (0) x 0 x 2x 2(x 0) 2+ + + + = = = = = , G ... ( ) 0,x G(x) G(0) G(x) G () x 0 x = = (1) ( )G (1) G(x) 0 x G () G (x) G (x) G(x) xG (x) x < < < < < 1 , 2 xG (x) G(x) F (x) 0 x = > ( ) [ )F 0, + 1 ) f 1 : y f (1)(x 1) f(1)= + f fC . f(x) f (1)x f (1) f(1) + (2) , ( )f x 0 f(x) 1 1 0 f (1) f (0) lim 0 x+ > > = 1 ( ) ( )x x lim f (1)x f (1) f(1) lim f (1)x + + + = = + (2) x lim f(x) + = + ( )f 0 t f(0) f(t) 1 x x 0 0 x 0 , dt f(t)dt x G(x)> , x lim x + = + x lim G(x) + = + , x x x x DLH G(x) lim F(x) lim lim G (x) lim f(x) x + + + + = = = + = , ) [ )x F(A) F(0), lim F(x) 1, + = = + ) 2 1 1 F (x) G(x) G (x), x 0 x x = + > 33. 33 0 2 2 G() G() G () G() G () G() G() G() G () > > > + > + 2 G() F () > (3) .. F ( ) 0, 2 F() F(0) G() F () 0 = = (3) 2 2 G() G() 0 > > 42 f f(2) 0 , f (x) 0 , x= < r ( )f "1 1" 2 x 4 x g(x) f (t)dt = ) x 4 x 0 0 g(x) f(t)dt f(t)dt , g (x) f (x) f(4 x) = = + g (x) f (x) f (4 x) = f ("1 1") g (x) 0 f (x) f (4 x) x 4 x x 2 = = = = ) ( )f g (x) 0 f (x) f (4 x) x 4 x x 2 , g (x) 0 x 2 > > < < < >L 2 g x 2= g(2) 0= ) 3 4 1 0 2 f(t)dt f(t)dt 2g(3) g(4) g(3) g(2) g(4) g(3)> > > ... g ( ) ( )1 2 2,3 , 3,4 1 2g ( ) g(3) g(2) g ( ) g(4) g(3) = = ( ) ( ) g ( ) 1 2 1 22 3 4 g g g(3) g(2) g(4) g(3) 2g(3) g(4) < < < < > > > 2 ) ( ) x 1 x 1 x e x 1 4 x e f(t)dt 0 g x e g(2) + > + > ) [ ) ( ]g 2, , x 2 g (x) g (2) g (x) 0,g ,2 , x 2 g (x) g (2) g (x) 0 + > < < < < )1 r x 1 x 1 x e 2 x 1 e 1 h(x 1) h(0) x 1 0 x 1 + < + < < < < 43 f f(1) 1= , r ( ) 2 x x t 1 1 1 2 z 5i f(t)dt z 5i e dt 12 x 1 ,x ,z + + + r c ) ( ) 2 x x t 1 1 1 g(x) 2 z 5i f(t)dt z 5i e dt 12 x 1 ,x = + + r g(x) 0 g(1) = . g x 1= 2 x 1 g (x) 2 z 5i f(x) z 5i 2xe 12 = + + . Fermat g (1) 0 2 z 5i 2 z 5i 12 0 = + + = 34. 34 M(z),E(5i) E ( 5i) z 5i z 5i 6 ME ME 6 + = = d 2 6 3 , 5 , 4= = = = = > . 2 2 y x 1 y 3 9 16 = ) 2 2 2x 3 y 1 9 y h(x) x 16 , x , y 3 16 4 = + = = + r () ( ) ( ) ( )22 23 9 A(1,0) , M x, x 16 , d A,M x 1 x 16 , x 4 16 + = + + r , ( ) ( ) ( ) 2 29 9 16 k(x) x 1 x 16 , x , k (x) 2 x 1 2x x 16 16 25 = + + = + =Lr k d(A,M) 16 x 25 = 16 3 641 M , 25 25 () hC A(1,0) . 0 0 0 0 9x y 1 1 16y x 1 = = ( )0 0 0 16 9x 16 x 1 x 25 = = ) ( )2 2 2 3 16 h (x) , h (x) 0 4 x 16 x 16 x 16 = = > + + + , h , (), 2 3 2 3x 3 2 h (x) 8 84 x 16 = = + x 0 2 x 2 x 4 2x 16 > = = + L , B(4,3 2) ( ) 2 2 3 2 4 3 2 8 12 41 d d(,) 41 3 2 1 8 = = = = + L 35. 35 ) ( ) x 1 (x) h(t)dt , H (x) h(x) 0 ,H ,x 1 H(x) H(1) 0= = > < < = 1 , ( ) [ ] 1 1 1 1 1 2 0 0 0 0 0 3 E H(x)dx x H(x)dx xH(x) xh(x)dx H(1) x x 16dx 4 = = = + = + + = 2 17 u x 16 2 2udu 2xdx 4 3 17 17 u du 16 4 4 = + = = = = L 44 f [ )0,+ ( ) 4 2x e 0 e x 2t 3 f(t)dt x dt , x 0 2t ln t + = + (1) ) 4 u ln te 4 4 1e 1dt du t 1 du dt u 1 2t ln t 2 u = = = = = (1) ( ) x 2 0 2t 3 f(t)dt x x+ = + ( ) 2x 1 2x 3 f(x) 1 2x f(x) 2x 3 + + = + = + ) [ ] 1 1 1 2x 1 2 E() dx 1 dx x ln(2x 3) ln(2 3) 1 ln5 2x 3 2x 3 + = = = + = + + + + ) ( ) ln(2 3) lim ln(2 3) lim 1 + + + + = = + DLH ln(2 3) 2 lim lim 0 2 3 + + + = = + lim() + = + ) t1 x 0 h(x) f(t)e dt= , x t 1 t 1f (t) 0ex 0 0 x x x x t 1 0 t 1 0 e e e f(t) f(t)e f(t)e x x >> 1 , t 1 11 1 1 1 1 x x x 0 0 0 0 0 f(t)dt f (t)e dt f (t)e dt f(t)dt h(x) e f(t)dt (1) [ ] 1 u11 x1 ux 0 x u 0 u 00 3 f(t)dt t ln(2t 3) 1 ln , lim e lim e 1 5 + + = + = + = + = = , (1) 1 x 3 3 1 ln h(x) e 1 ln 5 5 + + x 3 lim h(x) 1 ln 5+ = + 45 f [ ]1,9 ( ) ( ) ( ) 3 3 9 2 2 2 1 1 1 62 f t dt 2 f t dt 2 f t dt 3 + = ) ( ) ( ) 333 3 2 2 2 1 1 1 4t 62 2t 1 dt 4t 4t 1 dt 2t t 3 3 = + = + = ) [ ] [ ] [ ]2 t 1,9 t 1,9 t 1,3 , 2 9 3 3 u t u t 2 2 dt 2udu 1 1 1 f(t)dt 2 uf(u )dt 2 tf(t )dt= = = = , ( ) ( ) ( ) ( ) ( ) ( )( ) 3 3 3 3 3 222 2 2 2 2 1 1 1 1 1 f t dt 2 f t dt 4 tf t dt 2t 1 dt f t 2t 1 dt 0+ = = ( ) ( )( ) 2 2 g(t) f t 2t 1 0= 3 1 g(t)dt 0= 2 g(t) 0 f(t ) 2t 1= = 36. 36 t 0 2 t x t x > = = [ ] 1 1 f(x) 2 x 1 ,x 1,9 , f (x) , f (x) 0 x 2x x = = = < , f ) [ ]h(x) f(x)x 1 0 , x 1,9= , 2 2 1 x 1 1 1 1 h (x) 0 x , 1 3 1 9 1 3 x = = = < < > > > > h 2 1 x = 2 1 1 2 h = , . 1 2 h(x) 01 2 1 0 2 > , 1 2 = ) 0 0 1 f (x ) x 4 2 = = : 1 y f(4) f (4)(x 4) y x 1 2 = = + , f (4,3) , . fC : x 2y 7 0 / / + = , ( ) 22 4 2 3 7 5 d d(M,) 5 51 2 + = = = = + 46 f r f(0) 1= 2x 2 x e t 0 1 g(x) e dt ln tdt,x 0= + x h(x) e x= + ) [ ] 1 f (x) x x x xu f (t) f (u) t1 dt f (u)du 0 1 0 0 0 f (t)dt uf (u)du uf(u) f(u)du xf(x) f(u)du = = = = = x x 0 0 F(x) f(t)dt xf(x) f(u)du xf(x) 0= + = ( )1 x F (x) f(x) f f(x) f (x) f(x) xf (x) 0 = + = 14243 ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 x x x x x x x g (x) e ln e e x e x e xe g(x) xe c,g(0) 0 = + = + = = + = c 0= 2 x g(x) xe= ( ) 2 t 1 f(t) e , f (t) ln t = = ) ) ( )x h (x) e 1 0 , h = + > 1 1 h D h(A) = = r 1 1 1 e 1 1 1 u h (x) h(u) x ,dx h (u)du1 u h (1) u h(u) 1 u 0, 1 0 0 h (e 1) u h(u) e 1 u 1 3 E h (x) dx u h (u)du (ue u)du 2 + = = = = = = + = = + = = = + = = L ) : ( ) ( )1 1 y h (e 1) h e 1 (x e 1) + = + ( )( )1 h h x x = , ( )( )( ) ( ) ( )1 x e 11 1 1 1 h (e 1) 1 1 h h x h (x) 1 h (1) h (e 1) 1 h (e 1) e 1 = + + = = + = + = + : 1 y x e 1 = + 47 f r x u x 0 1 g(x) f(t)dt du e 1 0 , x = + r ) u 1 h(u) f(t)dt= x x 0 g(x) h(u)du e 1 0 g(0)= + = . g x 0= x g (x) h(x) e = + , . Fermat 1 0 g (0) 0 f(t)dt 1 = =
