Post on 27-Jun-2015
description
9 786188 021495
ISBN: 978-618-80214-9-5
nȍȨȦ ȰȪȵȦȨ�ȩȨȮ�ȚȯȢȞȪȷȦȞȭȞ�ȭȷȪȚ� ȜȢȚ�ȦȚ ȣȚȭȚȦȨȗȬȞȭȞ�ȭȢȫ�țȚȬȢȣȖȫ�ȖȦȦȨȢȞȫ� ȣȚȢ�ȦȚ�ȞȥțȚȡȶȦȞȭȞ�ȬȭȨȦ�ȭȪȵȩȨ�ȬȣȖȱȠȫ� ȣȚȢ ȞȪȜȚȬȘȚȫ��ȡȚ�ȭȨȦ�ȣȞȪȝȘȬȞȭȞ� ȩȨȤȤȚȩȤȕȬȢȨ�ȚȪȜȵȭȞȪȚ.»
ȁȞȵȝȲȪȨȫ�Ȇ��ȃȚȟȚȦȭȟȗȫ
ƲƳƱƭƱƥƱƴ
ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ� ȞĮ� ĮʌȠIJİȜȑıȠȣȞ� ȝȑȡȠȢ� IJȠȣȜȐȤȚıIJȠȞ� IJȘȢ�ĮʌȐȞIJȘıȘȢ�ıIJȠȞ�ȚįİȫįȘ�țȩıȝȠ�IJȠȣ��ȥȑȝĮIJȠȢ�ȩʌȠȣ�IJĮ�ʌȐȞIJĮ�įȘȜȫȞȠȣȞ�țĮȚ�įȘȜȫ�ȞȠȞIJĮȚ�ȦȢ�ĮȜȘșȚȞȐ��
ĬĮ�ȝʌȠȡȠȪıĮȞ� IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�ĮʌȠIJİȜȑıȠȣȞ� IJȘȞ�țĮȪıȚȝȘ�ȪȜȘ�Ș�ȠʌȠȓĮ�șĮ�İȞİȡȖȠʌȠȚȠȪıİ�IJȠȞ�ȞĮȡțȦȝȑȞȠ�ȖȓȖĮȞIJĮ�IJȘȢ�ıȣȞİȓįȘıȘȢ��
ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ�� ȦȢ� ȖȜȫııĮ� IJȘȢ� ȜȠȖȚțȒȢ�� ȞĮ� ıȣȞįȡȐȝȠȣȞ�ĮʌȠijĮıȚıIJȚțȐ�IJȚȢ�ıIJȡȚȝȦȖȝȑȞİȢ�įȣȞȐȝİȚȢ�IJȘȢ�ıȪȞİıȘȢ��IJȘȢ�IJĮʌİȚȞȩIJȘIJĮȢ��IJȘȢ�ȖȞȫ�ıȘȢ�țĮȚ�IJȘȢ�ĮȜȒșİȚĮȢ��Įʌȩ�IJĮ�ȝĮȞȚĮıȝȑȞĮ�ıIJȓijȘ�IJȠȣ�©șȑȜȦª��IJȠȣ�©ijĮȓȞȠȝĮȚª��IJȠȣ�©ȑȤȦª��IJȠȣ�©ȖȡȒȖȠȡĮª��IJȠȣ�©țĮIJĮȞĮȜȫȞȦª��IJȠȣ�©İȞIJȣʌȦıȚĮțȠȪª��IJȠȣ�©İȖȫª�
ȅ�țĮșȑȞĮȢ�Įʌ¶�ȩıȠȣȢ�ȕȡȓıțȠȣȞ�ȗȦȠȖȩȞȠ�IJȠ�ȞĮ�ĮıȤȠȜȠȪȞIJĮȚ�ıİ�ȕȐșȠȢ�ȝİ�IJȘ�ȗȦȒ��șĮ�ȑįȚȞİ�IJȘ�įȚțȒ�IJȠȣ�ĮʌȐȞIJȘıȘ�
ǻȪıțȠȜĮ�ʌȐȞIJȦȢ�șĮ�ȣʌȒȡȤĮȞ�įȚĮijȦȞȓİȢ��ıIJȠ�ȩIJȚ�șĮ�ȒIJĮȞ��ĮįȪȞĮIJȠȞ��ĮțȩȝȘ�țĮȚ�ȖȚĮ�IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�İʌȚIJİȜȑıȠȣȞ�ȑȡȖȠ�ıİ�țȐʌȠȚȠ�Įʌȩ�IJĮ�ʌȡȠȘȖȠȪȝİȞĮ��ĮȞ�țȐIJȚ�IJĮ�ĮʌȑțȠʌIJİ�Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȡȠijȩ�IJȠȣȢ��Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȠȣ�ĮʌİȓȡȠȣ�
ǻİȞ�ȞȠȝȓȗȦ�ȩIJȚ�ȤȡİȚȐȗİIJĮȚ�Ș�ıȣȞįȡȠȝȒ�IJȘȢ�țȕĮȞIJȠȝȘȤĮȞȚțȒȢ�� IJȘȢ�İȟİȜȚțIJȚ�țȒȢ�ȕȚȠȜȠȖȓĮȢ�Ȓ�IJȘȢ�șİȦȡȓĮȢ�IJȠȣ�ȤȐȠȣȢ��ȫıIJİ�ȞĮ ȠȡȚıIJİȓ�ȦȢ ȜȠȖȚțȒ�ĮȞĮțȠȜȠȣșȓĮ���Ș�ıȪȞįİıȘ�IJȠȣ�ĮʌİȓȡȠȣ�IJȦȞ�įȣȞĮIJȫȞ�ʌİȡȚʌIJȫıİȦȞ��ȝİ�IJȘȞ�IJȣȤȠįȚȦțIJȚțȠȪ�IJȪʌȠȣ��İʌȚȜȠȖȒ�țȐʌȠȚȦȞ�Įʌȩ�ĮȣIJȑȢ�
Ȃ¶�ĮȣIJȒ� ĮțȡȚȕȫȢ� IJȘȞ� ʌİʌȠȓșȘıȘ� ĮʌȑijȣȖĮ� İʌȚȝİȜȫȢ�� țȐșİ� ȠȜȠțȜȘȡȦIJȚțȠȪ�IJȪʌȠȣ�ȠȝĮįȠʌȠȓȘıȘ�IJȦȞ�ĮıțȒıİȦȞ��ǹȣIJȩ�įȘȜĮįȒ�ʌȠȣ�IJȩıȠ�İȪıIJȠȤĮ�Ƞ�ĮȟȑȤĮıIJȠȢ�įȐıțĮȜȠȢ�ĬİȩįȦȡȠȢ�ȀĮȗĮȞIJȗȒȢ�ȤĮȡĮțIJȒȡȚȗİ�ȦȢ�©ʌİȡȚʌIJȦıȚȠȜȠȖȓĮª�
ǻİȞ�ȑȤİȚ�ȞȩȘȝĮ�ȞĮ�ĮȞĮȗȘIJȒıİȚ�țĮȞİȓȢ�IJȚȢ�ĮȚIJȓİȢ�ĮȣIJȒȢ�IJȘȢ�İȝȝȠȞȒȢ��İȐȞ�įȘ�ȜĮįȒ�ȠijİȓȜİIJĮȚ�ıİ� ȑȜȜİȚȝȝĮ�ȜȠȖȚțȒȢ�Ȓ� İȞIJȚȝȩIJȘIJĮȢ��ȆȐȞIJȦȢ�� ȝȠȣ� İȓȞĮȚ� ȚįȚĮȓIJİȡĮ�įȪıțȠȜȠ�ȞĮ�ĮʌȠijȪȖȦ�IJȠȞ�ʌİȚȡĮıȝȩ�țȐʌȠȚȦȞ�İȞȠȤȜȘIJȚțȫȞ�ĮȜȜȐ�ȝȐȜȜȠȞ�ʌȡĮȖȝĮIJȚ�țȫȞ� ıțȑȥİȦȞ��ȦȢ� ʌȡȠȢ� IJĮ� ĮʌȠIJİȜȑıȝĮIJĮ� ĮȣIJȫȞ� IJȦȞ� ĮȞIJȚȜȒȥİȦȞ� Ȓ� ȓıȦȢ� “ȠȡijĮ�ȞȫȞ” ʌȡĮțIJȚțȫȞ�
ȀȐșİ�ȥİȣįĮȓıșȘıȘ�țĮȚ�ȝȚĮ�ijȣȜĮțȒ���ȀȐʌȦȢ�ȑIJıȚ�țȚ�İįȫ�Ș�ȥİȣįĮȓıșȘıȘ�IJȘȢ�ıȣıIJȘȝĮIJȚțȒȢ�ȠȡȖȐȞȦıȘȢ�țĮȚ�ȝİȜȑIJȘȢ�ijȣȜĮțȓȗİȚ�IJȠ�įȡȠıİȡȩ�ʌȞİȪȝĮ�IJȦȞ�ȝĮșȘIJȫȞ��ıIJİȡȫȞIJĮȢ�Įʌ¶�ĮȣIJȠȪȢ�IJȘ�ȝĮșȘIJİȓĮ�IJȘȢ�İȟİȡİȪȞȘıȘȢ��IJȘȞ�ĮȞİȝİȜȚȐ�IJȘȢ�ʌİȡȚʌȜȐȞȘ�ıȘȢ�țĮȚ�IJȘ�ȤĮȡȐ�IJȘȢ�ĮȞĮțȐȜȣȥȘȢ��ȈIJİȡȫȞIJĮȢ�ijȣıȚțȐ�ıIJȘ�ıȣȞȑȤİȚĮ�ıIJȠ�ȕĮșȝȩ�ʌȠȣ�IJĮ�İʌȘȡİȐȗȠȣȞ��IJĮ�ȝĮșȘȝĮIJȚțȐ�ĮȜȜȐ�țȣȡȓȦȢ�IJȘȞ�țȠȚȞȦȞȓĮ��Įʌȩ�ȝȣĮȜȐ�İȟİȡİȣȞȘȝȑ�ȞĮ�țĮȚ�ȐȡĮ�İȟİȡİȣȞȘIJȚțȐ��ȝİ� IJİȜȚțȩ�ĮʌȠIJȑȜİıȝĮ�IJȘȞ�ȤȦȡȓȢ�ĮȞIJȓʌĮȜȠ�İʌȚțȡȐIJȘıȘ�IJȘȢ�ȝȓȝȘıȘȢ��IJȘȢ�ijĮȚįȡȩIJȘIJĮȢ�țĮȚ�IJȘȢ�İʌĮȞȐȜȘȥȘȢ��ȅȚ�ĮʌȠįİȓȟİȚȢ�ȩȜȦȞ�IJȦȞ�ʌȡȠȘ�ȖȠȪȝİȞȦȞ��ʌȡȠijĮȞȫȢ�įİȞ�ȤȡİȚȐȗİIJĮȚ�ȞĮ�ĮȞĮȗȘIJȘșȠȪȞ ıİ�ıțȠȞȚıȝȑȞĮ�ȝĮșȘȝĮIJȚțȐ�ȕȚȕȜȓĮ��ǹȞ�IJİȜȚțȐ�ijȠȕȓȗİȚ�IJȩıȠ�ʌȠȜȪ�ȝȚĮ�ȝĮIJȚȐ�ȝȑıĮ�ȝĮȢ��IJȠȣȜȐȤȚıIJȠȞ�IJȫȡĮ�İȓȞĮȚ�ĮȡțİIJȒ�ȝȚĮ�ȝĮIJȚȐ�įȓʌȜĮ�ȝĮȢ�
ĬİııĮȜȠȞȓțȘ��ǿȠȪȞȚȠȢ 2013 īȚȫȡȖȠȢ ȁ��ȂĮȣȡȓįȘȢ
ƲƧƳƫƧƸƱƮƧƯƣ ƣƘ�ƵƱƮƱƶ
ȆȡȩȜȠȖȠȢ�
ȂȚȖĮįȚțȠȓ�ĮȡȚșȝȠȓ ........................................................................................ 11 � ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 113 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 141 ǵȡȚȠ�–�ȈȣȞȑȤİȚĮ�ıȣȞȐȡIJȘıȘȢ ....................................................................... 151� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 153 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 347 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 391 �ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��ǹǯ�ȂȑȡȠȢ� ............................................................... 401 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 403 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 477 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 501 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 507 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 569
ƲƧƳƫƧƸƱƮƧƯƣ ƤƘ�ƵƱƮƱƶ
ȆȡȩȜȠȖȠȢ� ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��Ǻǯ�ȂȑȡȠȢ� ............................................................... 11� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 259 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 317 ȅȜȠțȜȘȡȦIJȚțȩȢ�ȁȠȖȚıȝȩȢ ........................................................................... 329 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 331 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 505 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 543 ǼʌĮȞĮȜȘʌIJȚțȐ�ĬȑȝĮIJĮ ................................................................................ 555 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 575 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 636
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 243�
ƪƾǍǂ�����
DzıIJȦ� ıȣȞȐȡIJȘıȘ� f : o\ \ � Ș� ȠʌȠȓĮ� İȓȞĮȚ� įȪȠ� ijȠȡȑȢ� ʌĮȡĮȖȦȖȓıȚȝȘ������������ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�ȖȚĮ�țȐșİ� x�\ �ȚıȤȪȠȣȞ�ȠȚ�ıȤȑıİȚȢ:�
i)� � � � � � �h 0lim f x 2h 2f x h f x 0o
ª º� � � � ¬ ¼ ���
ii)�� � � � � �
h 0
f x 2h f x hlim f x
ho
c c� � �cc � � � � � � �
iii)�� � � � � � � �2h 0
f x 2h 2f x h f xOLP I [ �
ho
� � � �cc � �
�
��
�
ƭǞǔLj��i)� H ıȣȞȐȡIJȘıȘ�f ȦȢ�ʌĮȡĮȖȦȖȓıȚȝȘ�İȓȞĮȚ�țĮȚ�ıȣȞİȤȒȢ��DZȡĮ�
� � � � � � � � � � � �h 0OLP I [ �K �I [ K I [ I [ �I [ I [ ��o
� � � � � � ª º¬ ¼
�
ii)� DzȤȠȣȝİ
� � � � � � � � � � � �h 0 h 0
f x 2h f x h f x 2h f x f x h f xlim lim
h h ho o
c c c c c c� � � � � � �§ · �¨ ¸
© ¹
� � � � � � � �h 0 h 0
f x 2h f x f x h f x2lim lim
2h ho o
c c c c� � � � �
� � � � � �2f x f x f xcc cc cc � �
iii)� EijĮȡȝȩȗȠȞIJĮȢ�IJȠȞ�țĮȞȩȞĮ�De l’ Hospital ȑȤȠȣȝİ
� � � � � � � � � � � � � �� �� �
00
2h 0 h 0 2
f x 2h 2f x h f xf x 2h 2f x h f xlim lim
h ho o
c� � � �� � � �
c
� � � �h 0
2f x 2h 2f x hlim
2ho
c c� � �
� � � �h 0
f x 2h f x hlim
ho
c c� � �
� �ii)
f xcc �
244� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ������
DzıIJȦ� ıȣȞȐȡIJȘıȘ� f : o\ \ � Ș� ȠʌȠȓĮ� İȓȞĮȚ� įȪȠ� ijȠȡȑȢ� ʌĮȡĮȖȦȖȓıȚȝȘ� țĮȚ�țȣȡIJȒ��ĬİȦȡȠȪȝİ�țĮȚ�IJȘ�ıȣȞȐȡIJȘıȘ�g : o\ \ ��ȝİ�IJȪʌȠ�
� � � �g x f x 2x ȖȚĮ țȐșİ [ �� � � �ǹȞ� Ș� İȣșİȓĮ� ȝİ� İȟȓıȦıȘ� � y 2x � İȓȞĮȚ� ĮıȪȝʌIJȦIJȘ� IJȘȢ� fC � ıIJȠ� �f � țĮȚ���������ȣʌȐȡȤİȚ�IJȠ� � �
xlim f xo�f
c �ȞĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ���
i)� �Ș�ıȣȞȐȡIJȘıȘ�g�İȓȞĮȚ�țȣȡIJȒ�ii)� � �
xlim f xo�f
�f �țĮȚ� � �xlim g x 0o�f
c �
iii)� � �g x 0c � �ȖȚĮ�țȐșİ� x �� �iv)� Ș� fC �ȕȡȓıțİIJĮȚ�ʌȐȞȦ�Įʌȩ�IJȘȞ�ĮıȪȝʌIJȦIJȒ�IJȘȢ� \ �[� �
��
ƭǞǔLj�
i)� DzȤȠȣȝİ�
� � � �g x f x 2c c � ȖȚĮ�țȐșİ�� [ ��\
ǼʌİȚįȒ�Ș�f İȓȞĮȚ�țȣȡIJȒ��ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ�Ș ıȣȞȐȡIJȘıȘ� f c İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ��ȅʌȩ�IJİ��ȖȚĮ�țȐșİ� 1 2 1 2[ � [ ȝİ [ [� �� ȑȤȠȣȝİ
� � � �� � � �� � � �
1 2
1 2
1 2
f x f x
f x 2 f x 2
g x g x
c c�
c c� � � �
c c� �
DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ� gc İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ��įȘȜĮįȒ Ș�ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ țȣȡ�IJȒ�
ii)� z� Ǿ�İȣșİȓĮ� y 2x İȓȞĮȚ�ĮıȪȝʌIJȦIJȘ�IJȘȢ� fC ıIJȠ� ��f įȘȜĮįȒ�ȚıȤȪİȚ�Ș�ıȤȑıȘ
� �� �xOLP I [ �[ ��o�f
�
ǼʌȠȝȑȞȦȢ�
� � � �� �x xlim f x lim f x 2x 2xo�f o�f
ª º � � �f¬ ¼ ���ĮijȠȪ�xOLP �[ �o�f
�f
ȂǼĬȅǻȅȁȅīǿǹ�ǹʌȠįİȚțȞȪȠȣȝİ�ȩIJȚ Ș�ıȣȞȐȡ�IJȘıȘ� g İȓȞĮȚ� țȣȡIJȒ� ȝİ� IJȘ�ȕȠȒșİȚĮ� IJȠȣ� ȠȡȚıȝȠȪ��ǻȘȜĮ�įȒ�� ĮʌȠįİȚțȞȪȠȞIJĮȢ� ȩIJȚ� Ș�ıȣȞȐȡIJȘıȘ� gc İȓȞĮȚ� ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � ����
z� ǹʌȩ�IJĮ�ʌĮȡĮʌȐȞȦ�ʌĮȡĮIJȘȡȠȪȝİ�ȩIJȚ�IJȠ�� �
x
f xlim
xo�f İȓȞĮȚ�IJȘȢ�ȝȠȡijȒȢ� f
f țĮȚ�
ȖȞȦȡȓȗȠȣȝİ�ȩIJȚ�ȣʌȐȡȤİȚ�IJȠ� � �xOLP I [ �o�f
c ȅʌȩIJİ��ıȪȝijȦȞĮ�ȝİ�IJȠȞ�țĮȞȩȞĮ�De
L’ Hospital ȑȤȠȣȝİ�
� � � �� �
� �x x x
f x f xlim lim lim f x
x xo�f o�f o�f
cc
c
ǻȘȜĮįȒ�
� �x
2 lim f xo�f
c
țĮȚ İʌȠȝȑȞȦȢ��
� � � � � �x x x xOLP J [ OLP I [ � OLP I [ OLP � � � ��o�f o�f o�f o�f
c c c � � � ª º¬ ¼
iii)� Ǿ ıȣȞȐȡIJȘıȘ� gc İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�țĮȚ�ıȣȞİȤȒȢ��ȦȢ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȅʌȩIJİ��ȑȤİȚ�ıȪȞȠȜȠ�IJȚȝȫȞ IJȠ�įȚȐıIJȘȝĮ�
� � � �� � � �� �x x xOLP J [ � OLP J [ OLP J [ � � �o�f o�f o�f
c c c
ǼʌȠȝȑȞȦȢ�
� �g x 0c � ȖȚĮ�țȐșİ� x�� �
iv)� ǹʌȩ� IJȠ� İȡȫIJȘȝĮ� iii) ʌȡȠțȪʌIJİȚ� ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ� ȖȞȘıȓȦȢ�ijșȓȞȠȣıĮ�ıIJȠ��� ǼʌȓıȘȢ��ĮȟȚȠʌȠȚȫȞIJĮȢ�IJȠȞ�ȠȡȚıȝȩ�IJȘȢ�ĮıȪȝʌIJȦIJȘȢ�IJȘȢ� fC ıIJȠ �f ʌĮȓȡȞȠȣȝİ
� �� � � �x xOLP I [ �[ � OLP J [ ��o�f o�f
� �
ȀĮȚ�İʌİȚįȒ�Ș�ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ�ıȣȞİȤȒȢ��ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ�IJȠ�ıȪȞȠȜȠ�IJȚȝȫȞ�IJȘȢ�İȓȞĮȚ�IJȠ�įȚȐıIJȘȝĮ
� � � �� � � �� �x x xOLP J [ � OLP J [ �� OLP J [ �o�f o�f o�f
ǼʌȠȝȑȞȦȢ�
� �g x 0 ȖȚĮ țȐșİ [! �\
� �f x 2x 0� � ! ȖȚĮ�țȐșİ� x�\
� �f x 2x� ! ȖȚĮ�țȐșİ�� x�\ �
ǻȘȜĮįȒ��Ș� fC ȕȡȓıțİIJĮȚ�ʌȐȞȦ�Įʌȩ�IJȘȞ�ĮıȪȝʌIJȦIJȒ�IJȘȢ� \ �[�
246� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ������
ǻȓȞİIJĮȚ�Ș�ıȣȞȐȡIJȘıȘ
� � � � � �11 xf x x , x 0, 1 1,� � � � f
țĮȚ�Ș�ıȣȞȐȡIJȘıȘ�
� � � �ȕg x ĮOQ[ �� [ ��x
� � � � f��
�ȝİ Į� ȕ� �
IJȑIJȠȚĮ��ȫıIJİ�� � � � � �g x g 1 0 ȖȚĮ țȐșİ [ �� � �t � �
i)� NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Į ȕ �� �ii)� NĮ�ȝİȜİIJȒıİIJİ�IJȘ�ıȣȞȐȡIJȘıȘ�g��ȦȢ�ʌȡȠȢ�IJȘ�ȝȠȞȠIJȠȞȓĮ�țĮȚ�IJĮ�ĮțȡȩIJĮIJĮ� iii)� NĮ�ȝİȜİIJȒıİIJİ�IJȘ�ıȣȞȐȡIJȘıȘ�f�ȦȢ�ʌȡȠȢ�IJȘ�ȝȠȞȠIJȠȞȓĮ��iv)� ȃĮ�ȕȡİȓIJİ�IJȠ�ʌȜȒșȠȢ�IJȦȞ�ȡȚȗȫȞ�IJȘȢ�İȟȓıȦıȘȢ� � �f x Ȝ �ȖȚĮ� IJȚȢ� įȚȐijȠ�
ȡİȢ�IJȚȝȑȢ�IJȠȣ� Ȝ ��\ ���
ƭǞǔLj��i)� DzȤȠȣȝİ
� � ȕg 1 0 Į OQ� � �1
� � �
įȘȜĮįȒ ȕ � � ȕ ��� �
ǼʌȓıȘȢ� � � � �g x g 1t ȖȚĮ�țȐșİ��� � �[ �� � ��
ǻȘȜĮįȒ��Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ�IJȠʌȚțȩ�İȜȐȤȚıIJȠ�ȖȚĮ� [ �� ȀĮȚ�İʌİȚįȒ�İȓȞĮȚ�ʌĮȡĮ�ȖȦȖȓıȚȝȘ�ȝİ
� �
� �2
ȕg x Į OQ [ �x
Į ȕ ȖȚĮ țȐșİ [ ��x x
c§ ·c � �¨ ¸© ¹
� � � f
Įʌȩ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Fermat ʌȡȠțȪʌIJİȚ�ȩIJȚ
� � 2Į ȕg 1 0 0 Į ȕ ��1 1
c � � � �
ǵȝȦȢ�� ȕ �� ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ Į ȕ ��
ȆǹȇǹȉǾȇǾȈǾ�ȋȡİȚĮȗȩȝĮıIJİ� įȪȠ� ȚıȩIJȘIJİȢ�ıIJȚȢ� ȠʌȠȓİȢ� İȝʌȜȑțȠȞIJĮȚ� ȠȚ�ȐȖȞȦıIJȠȚ� ĮȡȚșȝȠȓ� Į� țĮȚ� ȕ��Ǿ�ʌȡȫIJȘ�ȚıȩIJȘIJĮ�
� �g 1 0 �
įȓȞİIJĮȚ� ȐȝİıĮ� Ǿ� įİȪIJİȡȘ�ȚıȩIJȘIJĮ� ʌȡȠțȪʌIJİȚ� Įʌȩ� IJȘȞ�ʌȜȘȡȠijȠȡȓĮ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ� IJȠʌȚțȩ� İȜȐȤȚ�ıIJȠ�ȖȚĮ� x 1 ��DZȡĮ��ıȪȝijȦȞĮ�ȝİ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Fermat
� �J � ��c
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 247�
ii)� ǹȞIJȚțĮșȚıIJȫȞIJĮȢ�ıIJȠȞ�IJȪʌȠ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�g IJȚȢ�IJȚȝȑȢ�IJȦȞ�Į�țĮȚ�ȕ�ʌȠȣ�ȕȡȒțĮȝİ�ıIJȠ�İȡȫIJȘȝĮ�i��ȑȤȠȣȝİ
� � � �1J [ OQ [ [� [ ��x
� � � � f �
ȅʌȩIJİ�
� �
� �
2
2
1J [ OQ [ �x
1 1x xx 1 ȖȚĮ țȐșİ [ �� �x
c§ ·c � �¨ ¸© ¹
�
� � � f
ǼʌȠȝȑȞȦȢ�
� � � 2
x 1J [ � � [ � � [ ��x�c � � � �
� � � 2
x 1J [ � � [ � � [ ��x�c � � � � � � � �
� � � 2
x 1J [ � � [ � � [ ��x�c ! � ! � � ! � !
DZȡĮ�� Ș� ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ� ȖȞȘıȓȦȢ� ijșȓȞȠȣıĮ� ıIJȠ� įȚȐıIJȘȝĮ� � @�� � țĮȚ� ȖȞȘıȓȦȢ�
ĮȪȟȠȣıĮ�ıIJȠ�įȚȐıIJȘȝĮ� > ��� � f ���ǼʌȓıȘȢ��Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ�ȠȜȚțȩ�İȜȐ�ȤȚıIJȠ�ȝȩȞȠ�ȖȚĮ� x 1 IJȠ� � �J � ��
iii)� DzȤȠȣȝİ
� �1
1 xf x x 0� ! ȖȚĮ�țȐșİ�� � � � �[ �� � �� �� � �f ȅʌȩIJİ�
� �1
1 x� OQ [
1 x 1 xOQ [OQ [f x e e e� � � ȖȚĮ�țȐșİ� � � � �[ ��� �� �� � �f
x 0 1 �f
gc � �
g
0
248� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ǼʌȠȝȑȞȦȢ�
� �
� � � � � � � �� �
� � � �
� �
� �� �
� �� � � �
OQ [ OQ [
1 x 1 x
1
1 x
1
1 x
1
1 x
1
1 x
2
2
2
2
OQ [f x e e1 x
OQ [ � [ OQ [ � [x
1 x1 � [ OQ [ �xx
1 x1 � OQ [xx
1 x
g xx ȖȚĮ țȐșİ [ ��� �� �
1 x
� �
�
�
�
�
c c§ · § ·c �¨ ¸ ¨ ¸�© ¹© ¹c c� � �
��
� � � � �
�
� � �
�
� � � � f�
ǵȝȦȢ��ıIJȠ�İȡȫIJȘȝĮ�i) ĮʌȠįİȓȟĮȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ�ȠȜȚțȩ�İȜȐȤȚıIJȠ�ȝȩȞȠ�ȖȚĮ� x 1 IJȠ� � �J � �� ǻȘȜĮįȒ�
� � � �g x g 1 0! ȖȚĮ�țȐșİ�� � � � �[ ��� �� �� � �f
DZȡĮ�
� � � �� �
11 x
2
g xf x x 0
1 x�c � !
� ȖȚĮ�țȐșİ�� � � � �[ ��� �� �� � �f
ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ��f İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�ıİ�țĮșȑȞĮ�Įʌȩ�IJĮ�įȚĮıIJȒȝĮIJĮ� � ���� țĮȚ��� � ��� � f �
�iv)� ȈȪȝijȦȞĮ�ȝİ�IJȠ�İȡȫIJȘȝĮ�iii) țĮȚ�ȝİ�ȕȐıȘ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�
f İȓȞĮȚ�ıȣȞİȤȒȢ�ʌȡȑʌİȚ�ĮȡȤȚțȐ�ȞĮ�ȣʌȠȜȠȖȓıȠȣȝİ�IJĮ�ȩȡȚĮ� � � � � � � � �
x 0 xx 1 x 1OLPI [ � OLP I [ � OLP I [ țĮȚ OLP I [ �
� �o o�fo o
DzȤȠȣȝİ�ȜȠȚʌȩȞ
� x 0 x 0
OQ [ �OLP OLP OQ [1 x 1 xo o
�f� �
ĮijȠȪ
x 0
1lim 11 xo
�
țĮȚ����x 0OLP OQ [ �o
�f
ȂǼĬȅǻȅȁȅīǿǹ�ǺȡȓıțȠȣȝİ� IJȠ� ıȪ�ȞȠȜȠ� IJȚȝȫȞ� IJȘȢ�ıȣȞȐȡIJȘıȘȢ� f ıİ�țĮșȑȞĮ� Įʌȩ� IJĮ� įȚ�ĮıIJȒȝĮIJĮ� ȝȠȞȠIJȠ�ȞȓĮȢ�IJȘȢ�
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 249�
ȅʌȩIJİ��șȑIJȠȞIJĮȢ� OQ [u1 x
�
ȑȤȠȣȝİ
� �OQ [
u1 xx 0 x 0 uOLPI [ OLPH OLP H ���o o o�f
� � �� �
00
[ � /�+� [ � [ � [ �
1OQ [OQ [ �xlim lim lim lim 1
1 x 1 x1 x
§ ·¨ ¸© ¹
o o o o
c § · � �¨ ¸� �c © ¹�
ȅʌȩIJİ��șȑIJȠȞIJĮȢ� OQ [u1 x
�
ȑȤȠȣȝİ
� �OQ [
u 11 xx 1 x 1 u 1
1OLPI [ OLPH OLP H H �e
��o o o�
� � �� �[ /�+� [ [ [
1OQ [OQ [ �xlim lim lim lim 0
1 x 1 x1 x
f§ ·¨ ¸f© ¹
o�f o�f o�f o�f
c § · � ¨ ¸� �c © ¹�
ȅʌȩIJİ��
� �OQ [uOQ [ 1 x u 01 x
x x u 0OLP I [ OLP H OLPH H ��
�
�o�f o�f o
ǹʌȩ�IJĮ�ʌĮȡĮʌȐȞȦ�ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ
� �� � � � � �x 0 x 1
1I ��� OLPI [ � OLP I [ ��e�o o
§ ·§ · ¨ ¸ ¨ ¸© ¹ © ¹
țĮȚ
� �� � � � � �xx 1
1I �� OLP I [ � OLP I [ � � �e� o�fo
§ ·§ ·� f ¨ ¸ ¨ ¸© ¹ © ¹
ǼʌȠȝȑȞȦȢ�� Ș� İȟȓıȦıȘ � �f x Ȝ ȑȤİȚ�
ȝȠȞĮįȚțȒ�ȜȪıȘ�ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ
1 1Ȝ �� � �e e
§ · § ·� �¨ ¸ ¨ ¸© ¹ © ¹
İȞȫ�İȓȞĮȚ�ĮįȪȞĮIJȘ�ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ
� @ > �1Ȝ � � �� �e
½� �f � � �f® ¾¯ ¿
x
y
y 1
y Ȝ
O 1
1e
���� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ�����
ǻȓȞİIJĮȚ�Ș�ıȣȞȐȡIJȘıȘ�
� � 3 4 3 2I [ ��[ OQ[ �[ ��[ [ � [ � � � � ! �
i)� ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ� f cc �İȓȞĮȚ�ȖȞȘıȓȦȢ�ijșȓȞȠȣıĮ��ii)� ȃĮ�ȕȡİȓIJİ�IJȠ�ıȪȞȠȜȠ�IJȚȝȫȞ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ� I �cc � �iii)� NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ș�ȖȡĮijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�f�ȑȤİȚ�ĮțȡȚ�
ȕȫȢ�ȑȞĮ�ıȘȝİȓȠ�țĮȝʌȒȢ����
ƭǞǔLj��i)� DzȤȠȣȝİ
� � � �3 4 3 2
2 3 3 2
2 3 2
I [ ��[ OQ [ �[ ��[ [
1��[ OQ [ ��[ ��[ ��[ �[x
��[ OQ [ ��[ ��[ �[ ȖȚĮ țȐșİ [ �
cc � � �
� � � � �
� � � !
țĮȚ � � � �2 3 2
2 2
2
I [ ��[ OQ [ ��[ ��[ �[
1��[ OQ [ ��[ ��[ ��[ �x
��[ OQ [ ��[ � ȖȚĮ țȐșİ [ ��
ccc � � �
� � � � �
� � !
ȅʌȩIJİ� � � � �
� �
2I [ ��[ OQ [ ��[ �
1��OQ [ ��[ ��[ �x
��[ OQ [ ��[ ���� OQ [ [ � ȖȚĮ țȐșİ [ ��
cccc � �
� � � �
� �
� � !
īȞȦȡȓȗȠȣȝİ�ȩȝȦȢ�ȩIJȚ OQ [ [ �d � ȖȚĮ�țȐșİ�� x 0!
țĮȚ�ȩIJȚ�Ș�ȚıȩIJȘIJĮ�ȚıȤȪİȚ�ȝȩȞȠ�ȖȚĮ� [ �� ǼʌȠȝȑȞȦȢ� � � � �I [ �� OQ [ [ � �ccc � � � ȖȚĮ�țȐșİ� � � � �[ ��� ��� � �f
țĮȚ � �I � ��ccc DZȡĮ�� Ș� ıȣȞȐȡIJȘıȘ� � f cc İȓȞĮȚ� ȖȞȘıȓȦȢ�ijșȓȞȠȣıĮ�ıIJȠ�įȚȐıIJȘȝĮ� � ��� �� f
ȂǼĬȅǻȅȁȅīǿǹ�ǺȡȓıțȠȣȝİ� IJȘȞ� ʌĮ�ȡȐȖȦȖȠ� IJȘȢ� ıȣȞȐȡ�IJȘıȘȢ� f cc įȘȜĮįȒ�IJȘ� ıȣȞȐȡIJȘıȘ� f ccc țĮȚ� ʌȡȠıʌĮșȠȪȝİ�ȞĮ� ĮȞȚȤȞİȪıȠȣȝİ� IJȠ�ʌȡȩıȘȝȩ�IJȘȢ� ȆȇȅȉǹȈǾ�ǿıȤȪİȚ
OQ [ [ �d � ȖȚĮ� țȐșİ� [ ��! Ǿ�ȚıȩIJȘIJĮ� ȚıȤȪİȚ� ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ [ �� �
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � ����
ii)� Ǿ� ıȣȞȐȡIJȘıȘ� f cc İȓȞĮȚ� ıȣȞİȤȒȢ� țĮȚ� ȖȞȘıȓȦȢ�ijșȓȞȠȣıĮ�ıIJȠ�įȚȐıIJȘȝĮ� � ��� �� f ǼʌȠȝȑȞȦȢ�
� �� � � � � �� �x x 0I �� OLP I [ � OLPI [ �
o�f occ cc cc� f
ǵȝȦȢ� � � � �2
x x
2
x
OLP I [ OLP ��[ OQ [ ��[ �
OQ [ �lim x 72 36x x
o�f o�f
o�f
cc � �
§ · � �¨ ¸© ¹
�f
įȚȩIJȚ 2
xlim xo�f
�f țĮȚ����
� �� �
x x x
/�+� [
x
OQ [ � OQ [ �lim 72 36 72 lim 36 limx x x x
OQ [72 lim 36 0
x172 lim 36x
�� � �� ���
o�f o�f o�f
f§ ·¨ ¸f© ¹
o�f
o�f
§ ·� � � �¨ ¸© ¹
c � �
�
� � �
ǼʌȓıȘȢ� � � � �
� �
� �
2
x 0 x 02
x 0 x 0
2
x 0
/�+� [ �
x 02
x 0
OLPI [ OLP ��[ OQ [ ��[ �
OLP��[ OQ [ OLP��[ �
OQ [72lim 36 0 21x
OQ [72lim 0 2
1x
1x72lim 21x
��OLP [ � �� � � ��
o o
o o
o
f§ ·¨ ¸f© ¹
o
o
o
cc � �
� �
� � �
c � �
c§ ·¨ ¸© ¹
��
� � � �
ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ � �� � � �I �� � � �cc � f �f
ȂǼĬȅǻȅȁȅīǿǹ�Ǿ� ıȣȞȐȡIJȘıȘ� f cc İȓȞĮȚ ıȣȞİȤȒȢ� țĮȚ� ȖȞȘıȓȦȢ�������������ijșȓȞȠȣıĮ� ıIJȠ� įȚȐıIJȘȝĮ�� ���� f �� ȅʌȩIJİ�� � ȖȚĮ� ȞĮ�ȕȡȠȪȝİ� IJȠ� ıȪȞȠȜȠ� IJȚȝȫȞ�IJȘȢ� Įȡțİȓ� � ȞĮ� ȕȡȠȪȝİ� IJĮ�ȩȡȚĮ�
� �x 0limf xo
cc țĮȚ� � �xOLP I [ �o�f
cc �
� �x c
���� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
iii)� ǹʌȩ�IJȠ�İȡȫIJȘȝĮ�ii) ʌȡȠțȪʌIJİȚ�ȩIJȚ�Ƞ�ĮȡȚșȝȩȢ�����ĮȞȒțİȚ�ıIJȠ� ıȪȞȠȜȠ� IJȚȝȫȞ� IJȘȢ� ıȣȞȐȡIJȘıȘȢ� f cc �� � ǻȘȜĮįȒ������� ȣʌȐȡȤİȚ� � �0[ ��� �f IJȑIJȠȚȠȢ��ȫıIJİ
� �0I [ ��cc
ȀĮȚ� İʌİȚįȒ� Ș� ıȣȞȐȡIJȘıȘ� f cc İȓȞĮȚ� ȖȞȘıȓȦȢ� ijșȓȞȠȣıĮ�ȑȤȠȣȝİ�
� � � �0f x f x 0cc cc! ȖȚĮ�țȐșİ�� � �0[ �� [�
țĮȚ � � � �0f x f x 0cc cc� ȖȚĮ�țȐșİ�� � �0[ [ � �� �f
DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ��f İȓȞĮȚ�țȣȡIJȒ�ıIJȠ�įȚȐıIJȘȝĮ� � @0�� [ țĮȚ�țȠȓȜȘ�ıIJȠ�įȚȐıIJȘȝĮ�
> �0[ � �� f ǼʌİȚįȒ�Ș�ıȣȞȐȡIJȘıȘ f cc ȝȘįİȞȓȗİIJĮȚ�ȝȩȞȠ�ıIJȠ�ıȘȝİȓȠ� 0x țĮȚ�İțĮIJȑ�
ȡȦșİȞ�ĮȜȜȐȗİȚ�ʌȡȩıȘȝȠ��ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ� IJȠ�ıȘȝİȓȠ� � �� �0 00 [ � I [ İȓȞĮȚ� IJȠ�
ȝȠȞĮįȚțȩ�ıȘȝİȓȠ�țĮȝʌȒȢ�IJȘȢ� f& �
ȂǼĬȅǻȅȁȅīǿǹ�ǹȡțİȓ� ȞĮ� ĮʌȠįİȓ�ȟȠȣȝİ� ȩIJȚ� Ș� ıȣȞȐȡ�IJȘıȘ� f cc ȝȘįİȞȓȗİ�IJĮȚ� ıİ� ȑȞĮ� ȝȩȞȠ�ıȘȝİȓȠ� İțĮIJȑȡȦșİȞ�IJȠȣ� ȠʌȠȓȠȣ� ĮȜȜȐȗİȚ�ʌȡȩıȘȝȠ�
x 0 0x �f
f cc � �
f
Ȉ�Ȁ�
0
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � ����
ƪƾǍǂ������
ǻȓȞİIJĮȚ�Ș�ıȣȞȐȡIJȘıȘ�
� � 2I [ [ �[OQ[� [ � � ! ��
i)� ȃĮ�ȝİȜİIJȒıİIJİ�IJȘ�ıȣȞȐȡIJȘıȘ��f��ȦȢ�ʌȡȠȢ�IJȘȞ�țȣȡIJȩIJȘIJĮ�țĮȚ�ȞĮ�ȕȡİȓ�IJİ��ĮȞ�ijȣıȚțȐ�ȣʌȐȡȤȠȣȞ�� IJĮ�ıȘȝİȓĮ�țĮȝʌȒȢ�IJȘȢ�ȖȡĮijȚțȒȢ�IJȘȢ�ʌĮȡȐ�ıIJĮıȘȢ��
ii)� NĮ�ȝİȜİIJȒıİIJİ�IJȘ�ıȣȞȐȡIJȘıȘ�f��ȦȢ�ʌȡȠȢ�IJȘ�ȝȠȞȠIJȠȞȓĮ�țĮȚ�IJĮ�ĮțȡȩIJĮIJĮ��iii)� NĮ�ȕȡİȓIJİ�IJȠ�ıȪȞȠȜȠ�IJȚȝȫȞ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�f��iv)� ȃĮ�ȕȡİȓIJİ�IJȠ�ʌȜȒșȠȢ�IJȦȞ�ȡȚȗȫȞ�IJȘȢ�İȟȓıȦıȘȢ�
Ȝ�OQ[ [x
�
� ȖȚĮ�IJȚȢ�įȚȐijȠȡİȢ�IJȚȝȑȢ�IJȠȣ� Ȝ ��\ ��ƭǞǔLj��i)� DzȤȠȣȝİ
� � � �2 1I [ [ �[ OQ [ �[ �OQ [ �[x
cc � � �
�[ �OQ [ � � � ȖȚĮ�țȐșİ�� x 0! � ȅʌȩIJİ
� � � �I [ �[ �OQ [ �
2 2x 22 ȖȚĮ țȐșİ [ �x x
ccc � �
� � !
ǼȓȞĮȚ�ȜȠȚʌȩȞ
� � � 2x 2I [ � � �[ � � [ ��x�cc � � � �
� � �x 02x 2I [ � � �[ � � [ ��
x
!�cc � � � � � � � �
� � �x 02x 2I [ � � �[ � � [ ��
x
!�cc ! � ! � � ! � !
Ȉȋȅȁǿȅ�Ǿ� ıȣȞȐȡIJȘıȘ� f cc ȝĮȢ�ʌĮȡȑȤİȚ�ȩıȘ�ʌȜȘȡȠijȠȡȓĮ�ȤȡİȚĮȗȩȝĮıIJİ� ȖȚĮ� IJȘȞ�țȣȡIJȩIJȘIJĮ�țĮȚ�IJĮ�ıȘȝİȓĮ�țĮȝʌȒȢ� IJȘȢ� ȖȡĮijȚțȒȢ�ʌĮȡȐıIJĮıȘȢ� IJȘȢ� ıȣȞȐȡ�IJȘıȘȢ�f��
x 0 1 �f
f cc � �
f
Ȉ�Ȁ�
0
���� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ� f İȓȞĮȚ�țȠȓȜȘ�ıIJȠ�įȚȐıIJȘȝĮ� � @�� � țĮȚ�țȣȡIJȒ�ıIJȠ�įȚȐıIJȘȝĮ�
> ��� � f ��Ǿ� fC ȑȤİȚ�ĮțȡȚȕȫȢ�ȑȞĮ�ıȘȝİȓȠ�țĮȝʌȒȢ�IJȠ� � �� �0 �� I � ��įȘȜĮįȒ�IJȠ�ıȘ�
ȝİȓȠ� � �0 ��� �
ii)� ǹʌȩ�IJȠ�İȡȫIJȘȝĮ�i) ʌȡȠțȪʌIJİȚ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ� f c
İȓȞĮȚ� ȖȞȘıȓȦȢ� ijșȓȞȠȣıĮ� ıIJȠ� įȚȐıIJȘȝĮ� � @��� țĮȚ�
ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�ıIJȠ�įȚȐıIJȘȝĮ� > ��� �� f DZȡĮ��Ș�
ıȣȞȐȡIJȘıȘ� f c ʌĮȡȠȣıȚȐȗİȚ� ȠȜȚțȩ� İȜȐȤȚıIJȠ�ȝȩȞȠ�ȖȚĮ� x 1 IJȠ
� �I � � �OQ� � ��c � �
ǼʌȠȝȑȞȦȢ� � � � �f x f 1 0c c! ȖȚĮ��țȐșİ�� � � � �[ ��� �� �� � �f
DZȡĮ�� Ș� ıȣȞȐȡIJȘıȘ� f İȓȞĮȚ� ȖȞȘıȓȦȢ� ĮȪȟȠȣıĮ����������ıIJȠ�įȚȐıIJȘȝĮ� � ��� � f țĮȚ�ijȣıȚțȐ�įİȞ�ȑȤİȚ�Įțȡȩ�
IJĮIJĮ� ii)� Ǿ�ıȣȞȐȡIJȘıȘ��f İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�ıIJȠ�įȚȐıIJȘȝĮ� � ��� � f ��Ǽʌȓ�
ıȘȢ�
� � � �
� � � �
2
x 0 x 0
2 2
x 0 x 0 x 0
/�+� [ � [ � [ �2
OLPI [ OLP [ �[ OQ [
�OQ [OLP[ OLP�[ OQ [ � OLP 1x
12�OQ [ xlim lim lim 2x 011xx
o o
o o o
f§ ·¨ ¸f© ¹
o o o
�
� �
c �
c§ · �¨ ¸© ¹
țĮȚ
� � � �2
x x
2
x
OLP I [ OLP [ �[ OQ [
OQ [lim x 1 2x
o�f o�f
o�f
�
§ · � �f¨ ¸© ¹
įȚȩIJȚ 2
xlim xo�f
�f
ȂǼĬȅǻȅȁȅīǿǹ�ȉȠ� ʌȡȩıȘȝȠ� IJȘȢ� ıȣȞĮȡ�IJȘıȘȢ� f cc ȝĮȢ�İȞȘȝİȡȫȞİȚ�ȖȚĮ� IJȘȞ� țȣȡIJȩIJȘIJĮ� IJȘȢ�ıȣȞȐȡIJȘıȘȢ� f�� � ǻȘȜĮįȒ��ȖȚĮ� IJȘ� ȝȠȞȠIJȠȞȓĮ� IJȘȢ� ıȣ�ȞȐȡIJȘıȘȢ� f c �� Ǿ� ȖȞȫıȘ�IJȘȢ�ȝȠȞȠIJȠȞȓĮȢ� IJȘȢ� f c ĮȞ�ıȣȞįȣĮıIJİȓ�ȝİ�IJȘȞ�İȪȡİıȘ�IJȦȞ�ȡȚȗȫȞ�IJȘȢ�ȠįȘȖİȓ�ıIJȘȞ�ĮȞĮȖȞȫȡȚıȘ� IJȠȣ� ʌȡȠıȒ�ȝȠȣ� IJȘȢ� f c țĮȚ� ıȣȞİʌȫȢ�ıIJȘȞ� İȪȡİıȘ� IJȘȢ� ȝȠȞȠIJȠ�ȞȓĮȢ�IJȘȢ�f��
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � ����
țĮȚ
� �� �
x x
/�+� [
x
OQ [ OQ [lim 1 2 1 2 limx x
OQ [1 2 lim
x11 2 limx
� � � ��
o�f o�f
f§ ·¨ ¸f© ¹
o�f
o�f
§ ·� �¨ ¸© ¹
c �
c
�
� �
ǼʌȠȝȑȞȦȢ�
� �� � � � � �� � � �x 0 x
I �� OLPI [ � OLP I [ �� �o o�f
� f � f
iv) H İȟȓıȦıȘ
Ȝ�OQ [ [x
�
ȠȡȓȗİIJĮȚ� ȖȚĮ� x 0! țĮȚ� ȚıȠįȪȞĮȝĮ� ȖȡȐ�ijİIJĮȚ
2
2
2
�[ OQ [ [ Ȝ
[ �[ OQ [ Ȝ
[ �[ OQ [ Ȝ
�
� � � �
� �
� �f x Ȝ�� (1)
Ǿ�İȟȓıȦıȘ�����ȑȤİȚ�ȜȪıȘ�ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ
� �� � � �Ȝ I �� �� �� �f � f
ȀĮȚ� İʌİȚįȒ� Ș� ıȣȞȐȡIJȘıȘ� f İȓȞĮȚ� 1 1� ��ıȣȝʌİȡĮȓȞȠȣȝİ� ȩIJȚ� Ș� İȟȓıȦıȘ� (1) ȐȡĮ�țĮȚ� Ș� ȚıȠįȪȞĮȝȒ� IJȘȢ� įȠșİȓıĮ� İȟȓıȦıȘ�ȑȤİȚ�ȝȠȞĮįȚțȒ�ȡȓȗĮ�ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ�
� �Ȝ ��� �f
İȞȫ�İȓȞĮȚ�ĮįȪȞĮIJȘ�ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ� � @Ȝ � � �� �f
ȂǼĬȅǻȅȁȅīǿǹ�ȁȪȞȠȣȝİ�ȦȢ�ʌȡȠȢ�IJȘȞ�ʌĮȡȐȝİIJȡȠ�Ȝ� ȖȡȐijȠȞIJĮȢ� IJȘȞ� İȟȓıȦıȘ� ıIJȘ�ȝȠȡijȒ�
� �f x Ȝ� ȅʌȩIJİ�� IJȠ� ʌȡȩȕȜȘȝĮ� ĮȞȐȖİIJĮȚ�ıIJȘȞ� İȪȡİıȘ� IJȠȣ� ʌȜȒșȠȣȢ� IJȦȞ�țȠȚȞȫȞ� ıȘȝİȓȦȞ� IJȘȢ� fC ȝİ� IJȘȞ�İȣșİȓĮ� y Ȝ� DzIJıȚ�� ʌĮȡȩȜȠ� ʌȠȣ�ıIJȘȞ� İʌȓȜȣıȘ� IJȠȣ� ʌȡȠȕȜȒȝĮIJȠȢ�įİȞ� İʌȚțĮȜȠȪȝĮıIJİ� IJȘȞ� fC İȞ�IJȠȪIJȠȚȢ� Ș� ȝȠȡijȒ� IJȘȢ� İȓȞĮȚ� ĮȣIJȒ�ʌȠȣ�ȝĮȢ�İȝʌȞȑİȚ�țĮȚ�ȝĮȢ�ȠįȘȖİȓ���
x
y
fC
y Ȝ
O 1
1
���� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
ƪƾǍǂ�����
DzıIJȦ�ıȣȞȐȡIJȘıȘ� � �f : 1, � f o \ Ș�ȠʌȠȓĮ�İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ�țĮȚ�IJȑIJȠȚĮ��ȫıIJİ�� � �f e 1 �țĮȚ�
� �� �
� �� �
2 f xf xx 4 4x
f xf x
c�
c���ȖȚĮ�țȐșİ�� x 1! ��
i)� ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�� � 2I [ OQ [ ���ȖȚĮ�țȐșİ�� [ ��! �
ii)� NĮ� ȕȡİȓIJİ� IJȘȞ� İȟȓıȦıȘ� IJȘȢ� İijĮʌIJȠȝȑȞȘȢ� IJȘȢ� fC � ʌȠȣ� įȚȑȡȤİIJĮȚ� Įʌȩ�IJȘȞ�ĮȡȤȒ�IJȦȞ�ĮȟȩȞȦȞ��
iii)� ȃĮ�ȕȡİȓIJİ�IJȚȢ�ĮıȪȝʌIJȦIJİȢ�IJȘȢ� f& �c
iv)� ȃĮ�ȝİȜİIJȒıİIJİ�IJȘ�ıȣȞȐȡIJȘıȘ�f��ȦȢ�ʌȡȠȢ�IJȘȞ�țȣȡIJȩIJȘIJĮ�țĮȚ�ȞĮ�ȕȡİȓIJİ�IJĮ�ıȘȝİȓĮ�țĮȝʌȒȢ�IJȘȢ� f& � �
��
ƭǞǔLj��i)� Ǿ�įȠșİȓıĮ�ȚıȩIJȘIJĮ�ıȣȞĮȡIJȒıİȦȞ�ȚıȠįȪȞĮȝĮ�ȖȡȐijİIJĮȚ
� � � � � � � � � � � �2x f x f x 4 f x f x 4xf x f xc c c� ȖȚĮ�țȐșİ�� x 1!
� �� � � �� � � � � �222x f x 4 f x 4xf x f x 0c c� � � ȖȚĮ�țȐșİ�� x 1!
� �� � � �� � � �� � � �� �22xf x 2 f x 2 xf x 2 f x 0c c� � � ȖȚĮ�țȐșİ�� x 1!
� � � �� �2xf x 2 f x 0c� � ȖȚĮ�țȐșİ�� x 1!
ǻȘȜĮįȒ� � � � � xf x 2 f x 0c � ȖȚĮ�țȐșİ��� x 1!
� � � � xf x 2 f xc� ȖȚĮ�țȐșİ��� x 1!
� �� �
f x 1 x2 f x
c� ȖȚĮ�țȐșİ��� x 1!
� �� � � �� I [ OQ [c c� ȖȚĮ�țȐșİ��� x 1!
� �� I [ OQ [ F� � ȖȚĮ�țȐșİ��� x 1! ��ȩʌȠȣ� F ��\
ƦNJǂǗǐǒNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � ����
ĬȑIJȠȞIJĮȢ�ıIJȘȞ�IJİȜİȣIJĮȓĮ�ıȤȑıȘ�ȩʌȠȣ�x IJȠ�e ȑȤȠȣȝİ � �I H OQ H F � � F F �� � � � �
ǼʌȠȝȑȞȦȢ� � �I [ OQ [ ȖȚĮ�țȐșİ��� x 1!
țĮȚ�IJİȜȚțȐ � � 2I [ OQ [ ȖȚĮ�țȐșİ�� [ ��!
�
ii)� DzȤȠȣȝİ
� � � � � �2I [ OQ [ �OQ [ OQ [
� OQ [�OQ [ � ȖȚĮ țȐșİ [ ��x x
c cc
� !
DzıIJȦ�� � �� �0 00 [ � I [ IJȠ�ıȘȝİȓȠ�İʌĮijȒȢ�IJȘȢ�ȗȘIJȠȪȝİȞȘȢ�İȣșİȓĮȢ
� � � �� �0 0 0İ � \ I [ I [ [ [c� � � Ǿ� İȣșİȓĮ� � �İ įȚȑȡȤİIJĮȚ� � Įʌȩ� IJȘȞ� ĮȡȤȒ� IJȦȞ� ĮȟȩȞȦȞ�
� �ȅ �� � ĮȞ�țĮȚ�ȝȩȞȠ�ĮȞ�ȚıȤȪİȚ� � � � �� �0 0 00 f x f x 0 xc� � � � � �0 0 0x f x f x 0c� �
� �
200 0
0
20 0
0 0
0 0
0
20
OQ [[ � OQ [ �x
�OQ [ OQ [ �
OQ [ � OQ [ �
� OQ [ �� ĮijȠȪ OQ [ OQ� �
OQ [ �
[ H �
� �
� �
� �
� � !
�
�
ȅʌȩIJİ� � � � � � �22 2 2
0I [ I H OQ H �OQH � țĮȚ
� � � �2
20 2 2 2
OQ H � �I [ I H � � �e e e
c c
ǼʌȠȝȑȞȦȢ��Ș�İȣșİȓĮ� � �İ ȑȤİȚ�İȟȓıȦıȘ
� �22
4y 4 x ee
� �
įȘȜĮįȒ
2 24 4\ � [ � \ [�e e
� � �
�
ȂǼĬȅǻȅȁȅīǿǹ�ǼȪȡİıȘ� İijĮʌIJȩȝİ�ȞȘȢ� İȣșİȓĮȢ� ıȘ�ȝĮȓȞİȚ� İȪȡİıȘ� IJȠȣ�ıȘȝİȓȠȣ� İʌĮijȒȢ�
� �� �0 00 [ � I [ � �
���� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�
iii)� DzȤȠȣȝİ
� � � �OQ [I [ � � [ ��x
c � � f
Ǿ�ıȣȞȐȡIJȘıȘ� f c İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ�įȚȐıIJȘȝĮ� � ��� � f țĮȚ�ıȣȞİʌȫȢ�Ș�ȝȩȞȘ�ʌȚșĮ�ȞȒ�țĮIJĮțȩȡȣijȘ�ĮıȪȝʌIJȦIJȘ�IJȘȢ� fC c İȓȞĮȚ�Ș�İȣșİȓĮ� [ �� ǵȝȦȢ�
� �x 1 x 1
OQ [ OQ�OLPI [ OLP� � ��x 1o o
c
DZȡĮ��Ș� fC c įİȞ�ȑȤİȚ�țĮIJĮțȩȡȣijȘ�ĮıȪȝʌIJȦIJȘ��ȈIJȘ�ıȣȞȑȤİȚĮ�İȟİIJȐȗȠȣȝİ�ĮȞ�Ș� fC c ȑȤİȚ�ȠȡȚȗȩȞIJȚĮ�ĮıȪȝʌIJȦIJȘ�ıIJȠ� ��f DzȤȠȣȝİ
� � � �� �[ [ /�+� [ [ [
12�OQ [�OQ [ �xOLP I [ OLP OLP OLP OLP ��x 1 xx
f§ ·¨ ¸f© ¹
o�f o�f o�f o�f o�f
�cc
c
DZȡĮ�� Ș� İȣșİȓĮ� y 0 �� įȘȜĮįȒ� Ƞ� ȐȟȠȞĮȢ� x xc �� İȓȞĮȚ� Ș� ȝȠȞĮįȚțȒ� ȠȡȚȗȩȞIJȚĮ� ĮıȪ�ȝʌIJȦIJȘ�IJȘȢ� fC c ��ǼʌȓıȘȢ��ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ�Ș� fC c įİȞ�ȑȤİȚ�ʌȜȐȖȚĮ�ĮıȪȝʌIJȦIJȘ�
iv)� DzȤȠȣȝİ
� � � �� � � � � �
� �
2
2 2
OQ [ [ OQ [ [OQ [ OQ [f x f x 2 2 2x x x
1 [ OQ [ � OQ [x2 2 ȖȚĮ țȐșİ [ �� �x x
c cc c �§ · § ·ccc c ¨ ¸ ¨ ¸© ¹ © ¹
� � � � f
ǼʌȠȝȑȞȦȢ�
� � � � OQ [I [ � � � � OQ [ � OQ [ OQH [ Hx
�cc � � � � �
� � �x 1� OQ [I [ � � � � OQ [ � OQ [ OQH [ H
x
!�cc � � � � � � � ! � !
� � �x 1� OQ [I [ � � � � OQ [ � OQ [ OQH � [ H�
x
!�cc ! � ! � � ! � � � � �
ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ��f İȓȞĮȚ�țȣȡIJȒ�ıIJȠ�įȚȐıIJȘȝĮ� � @�� H țĮȚ�țȠȓȜȘ� ıIJȠ� įȚȐıIJȘȝĮ� > �H� �� f Ǿ� � fC ȑȤİȚ� ĮțȡȚȕȫȢ� ȑȞĮ� ıȘȝİȓȠ� țĮȝʌȒȢ� IJȠ�
� �� �$ H� I H ��įȘȜĮįȒ�IJȠ�ıȘȝİȓȠ� � �$ H�� �
x 1 e �f
f cc � �
f
Ȉ�Ȁ�
0