Question

ÒÑÖªº¯Êýf£¨x£©=e^x£¨x³-6x²+3x+a£©£¬
£¨¢ñ£©µ±a=1ʱ£¬Çóº¯ÊýÔÚ£¨0£¬f£¨0£©£©´¦µÄÇÐÏß·½³Ì£»
£¨¢ò£©Èôº¯Êýf£¨x£©ÓÐÈý¸ö¼«Öµµã£¬ÇóʵÊýaµÄÈ¡Öµ·¶Î§£»
£¨¢ó£©¶¨Ò壺Èç¹ûÇúÏßCÉÏ´æÔÚ²»Í¬µãµÄÁ½µãA£¨x₁£¬y₁ £©£¬B£¨x₂£¬y₂ £©£¬¹ýABµÄÖеãÇÒ´¹Ö±ÓÚxÖáµÄÖ±Ïß½»ÇúÏßCÓÚµãM£¬Ê¹µÃÖ±ÏßABÓëÇúÏßCÔÚM´¦µÄÇÐÏßƽÐУ¬Ôò³ÆÇúÏßCÓС°Æ½ºâÇÐÏß¡±£®
ÊÔÅжϺ¯ÊýG£¨x£©=[f'£¨x£©-f£¨x£©]•e^-x+e^xµÄͼÏóÊÇ·ñÓС°Æ½ºâÇÐÏß¡±£¬ÎªÊ²Ã´£¿

Answer 1

·ÖÎö£ºÏȶԺ¯ÊýÇóµ¼Êýf'£¨x£©=e^x£¨x³-3x²-9x+a+3£©£¬
£¨¢ñ£©µ±a=1ʱ£¬f'£¨0£©¼°f£¨0£©¾ù¿ÉÇ󣬽ø¶ø¿ÉµÃº¯ÊýÔÚ£¨0£¬f£¨0£©£©´¦µÄÇÐÏß·½³Ì£»
£¨¢ò£©ÓÉÓÚf£¨x£©ÓÐÈý¸ö¼«Öµµã?f'£¨x£©=e^x£¨x³-3x²-9x+a+3£©ÓÐÈý¸öÁãµã?g£¨x£©=x³-3x²-9x+a+3ÓÐÈý¸öÁãµã£¬
¼´ÒªÇóg£¨x£©µÄ¼«´óֵΪÕý£¬ÇÒ¼«Ð¡ÖµÎª¸º£¬Ôò¿ÉÇó³öʵÊýaµÄÈ¡Öµ·¶Î§£»
£¨¢ó£©ÏÈÅж¨º¯ÊýG£¨x£©µÄ½âÎöʽ£¬ÔÙÇó³öÇúÏßÔÚµãM´¦µÄÇÐÏßбÂʼ°Ö±ÏßABµÄбÂÊ£¬ÕûÀíºó¹¹½¨Ðº¯Êý£¬
½èÖúÓÚк¯ÊýµÄµ¥µ÷ÐÔÀ´ÅжϺ¯ÊýG£¨x£©=[f'£¨x£©-f£¨x£©]•e^-x+e^xµÄͼÏóÊÇ·ñÓС°Æ½ºâÇÐÏß¡±£®

½â´ð£º½â£ºf'£¨x£©=e^x£¨x³-3x²-9x+a+3£©
£¨¢ñ£©µ±a=1ʱf'£¨0£©=4£¬f£¨0£©=1
º¯ÊýÔÚ£¨0£¬f£¨0£©£©´¦µÄÇÐÏß·½³ÌΪy=4x+1
£¨¢ò£© f'£¨x£©=e^x£¨x³-3x²-9x+a+3£©
Éèg£¨x£©=x³-3x²-9x+a+3£¬Ôòg'£¨x£©=3x²-6x-9=3£¨x-3£©£¨x+1£©
¡àg£¨x£©µÄ¼«´óֵΪg£¨-1£©=a+8£¬¼«Ð¡ÖµÎªg£¨3£©=a-24£¬
ÓÉÓÚf£¨x£©ÓÐÈý¸ö¼«Öµµã?f'£¨x£©ÓÐÈý¸öÁãµã?g£¨x£©ÓÐÈý¸öÁãµã
¡àg£¨x£©µÄ¼«´óֵΪÕý£¬ÇÒ¼«Ð¡ÖµÎª¸º£¬¼´ a+8£¾0£¬a-24£¼0
¿ÉµÃ-8£¼a£¼24
£¨¢ó£©ÓÉÌâÒâÖª£¬G£¨x£©=[f'£¨x£©-f£¨x£©]e^-x+e^x=e^x+3x²-12x+3
¡àG'£¨x£©=e^x+6x-12
¹ÊG£¨x£©µÄͼÏóÔÚM´¦µÄÇÐÏßµÄбÂÊΪk0=G¡ä(

x1+x2

2

)=e

x1+x2

2

+3(x1+x2)-12
Ö±ÏßABµÄбÂÊkAB=

G(x1)-G(x2)

x1-x2

=

ex1-ex2

x1-x2

+3(x1+x2)-12
Èç¹ûk₀=k_AB£¬Ôò

ex1-ex2

x1-x2

=e

x1+x2

2

Ôò ex1-ex2=e

x1+x2

2

(x1-x2)¿É»¯Îªe

x1-x2

2

-e

x2-x1

2

=(x1-x2)
Áî

x1-x2

2

=t£¬ÉÏʽ¼´Îªe^t-e^-t=2t
¹¹Ô캯Êýh£¨x£©=e^x-e^-x-2x£¬Ôòh'£¨x£©=e^x+e^-x-2¡Ý0£¬Ôòh£¨x£©ÔÚRÉÏÊÇÔöº¯Êý£¬
ÒòΪh£¨0£©=0£¬ËùÒÔh£¨t£©=0µÄ³äÒªÌõ¼þÊÇt=0£®´Ëʱ x₁=x₂ÓëÌõ¼þì¶Ü£®
ËùÒÔG£¨x£©µÄͼÏóûÓС°Æ½ºâÇÐÏß¡±

µãÆÀ£º±¾Ì⿼²éº¯ÊýÓëµ¼ÊýµÄ×ÛºÏÓ¦Óã¬Éæ¼°¸ùµÄ¸öÊýµÄÅжϣ¬ÊôÖеµÌ⣮