Question

15£®ÒÑÖªf£¨x£©=mx-alnx-m£¬g£¨x£©=$\frac{ex}{e^x}$£¬ÆäÖÐm£¬a¾ùΪʵÊý£¬
£¨1£©Çóg£¨x£©µÄ¼«Öµ£»
£¨2£©Éèm=1£¬a=0£¬ÇóÖ¤¶Ô$?{x_1}£¬{x_2}¡Ê[{3£¬4}]£¨{x_1}¡Ù{x_2}£©£¬|{f£¨{x_2}£©-f£¨{x_1}£©}|£¼|{\frac{{e{x_2}}}{{g£¨{x_2}£©}}-\frac{{e{x_1}}}{{g£¨{x_1}£©}}}$|ºã³ÉÁ¢£»
£¨3£©Éèa=2£¬Èô¶Ô?¸ø¶¨µÄx₀¡Ê£¨0£¬e]£¬ÔÚÇø¼ä£¨0£¬e]ÉÏ×Ü´æÔÚt₁£¬t₂£¨t₁¡Ùt₂£©Ê¹µÃf£¨t₁£©=f£¨t₂£©=g£¨x₀£©³ÉÁ¢£¬ÇómµÄÈ¡Öµ·¶Î§£®

Answer 1

·ÖÎö £¨1£©Çó³öº¯ÊýµÄµ¼Êý£¬ÀûÓõ¼º¯ÊýµÄ·ûºÅÅжϺ¯ÊýµÄµ¥µ÷ÐÔ£¬È»ºóÇó½â¼«Öµ£®
£¨2£©Í¨¹ým=1£¬a=0£¬»¯¼òf£¨x£©=x-1£¬ÀûÓú¯ÊýµÄµ¥µ÷ÐÔ£¬×ª»¯Ô­²»µÈʽת»¯$f£¨{x_2}£©-f£¨{x_1}£©£¼\frac{{e{x_2}}}{{g£¨{x_2}£©}}-\frac{{e{x_1}}}{{g£¨{x_1}£©}}$£¬
¹¹Ô캯Êý$h£¨x£©=f£¨x£©-\frac{ex}{g£¨x£©}=x-{e^x}-1$£¬ÀûÓÃк¯ÊýµÄµ¼ÊýµÄµ¥µ÷ÐÔ£¬Ö¤²»µÈʽ³ÉÁ¢£®
£¨3£©ÓÉ£¨1£©µÃg£¨x£©µÄ×î´óÖµ£¬Çó³öº¯Êýf£¨x£©µÄµ¼Êý£¬ÅжÏm¡Ü0£¬²»Âú×ãÌâÒ⣻µ±m£¾0ʱ£¬Òª?t₁£¬t₂ʹµÃf£¨t₁£©=f£¨t₂£©£¬f£¨x£©µÄ¼«Öµµã±ØÔÚÇø¼ä£¨0£¬e£©ÄÚ£¬Çó³ömµÄ·¶Î§£¬µ±$m£¾\frac{2}{e}$£¬ÀûÓÃg£¨x£©ÔÚ£¨0£¬e£©ÉϵÄÖµÓò°üº¬ÓÚf£¨x£©ÔÚ$£¨{0£¬\frac{2}{m}}£©ºÍ£¨{\frac{2}{m}£¬e}£©$ÉϵÄÖµÓò£¬ÍƳö¹Øϵʽ£¬Í¨¹ý¹¹Ô캯Êýw£¨x£©=2e^x-x£¬Í¨¹ýµ¼ÊýÇó½âº¯ÊýµÄ×îÖµ£¬È»ºóÍƳö$m¡Ý\frac{3}{e-1}$£®

½â´ð ½â£º£¨1£©¡ß$g£¨x£©=\frac{ex}{e^x}$£¬¡à${g^'}£¨x£©=\frac{-e£¨x-1£©}{e^x}$£¬
¡à£¨-¡Þ£¬1£©¡ü£¬£¨1£¬+¡Þ£©¡ý£¬
¡àg£¨x£©¼«´óÖµg£¨1£©=1£¬ÎÞ¼«Ð¡Öµ£»¡­£¨4·Ö£©
£¨2£©¡ßm=1£¬a=0£¬¡àf£¨x£©=x-1£¬ÔÚ[3£¬4]ÉÏ ÊÇÔöº¯Êý¡à$\frac{ex}{g£¨x£©}={e^x}$£¬ÔÚ[3£¬4]ÉÏÊÇÔöº¯Êý
Éè3¡Üx₁£¼x₂¡Ü4£¬ÔòÔ­²»µÈʽת»¯Îª$f£¨{x_2}£©-f£¨{x_1}£©£¼\frac{{e{x_2}}}{{g£¨{x_2}£©}}-\frac{{e{x_1}}}{{g£¨{x_1}£©}}$
¼´$f£¨{x_2}£©-\frac{{e{x_2}}}{{g£¨{x_2}£©}}£¼f£¨{x_1}£©-\frac{{e{x_1}}}{{g£¨{x_1}£©}}$¡­£¨6·Ö£©
Áî$h£¨x£©=f£¨x£©-\frac{ex}{g£¨x£©}=x-{e^x}-1$£¬
¼´Ö¤?x₁£¼x₂£¬h£¨x₂£©£¼h£¨x₁£©£¬¼´h£¨x£©ÔÚ[3£¬4]¡ý
¡ßh¡ä£¨x£©=1-e^x£¼0ÔÚ[3£¬4]ºã³ÉÁ¢¼´h£¨x£©ÔÚ[3£¬4]¡ý£¬
¼´ËùÖ¤²»µÈʽ³ÉÁ¢£®¡­£¨9·Ö£©
£¨3£©ÓÉ£¨1£©µÃg£¨x£©ÔÚ£¨0£¬1£©¡ü£¨1£¬e£©¡ý£¬g£¨x£©_max=g£¨1£©=1
ËùÒÔ£¬g£¨x£©¡Ê£¨0£¬1]
ÓÖ${f^'}£¨x£©=m-\frac{2}{x}£¬µ±m¡Ü0ʱ£¬{f^'}£¨x£©£¼0£¬f£¨x£©ÔÚ£¨{0£¬e}£©¡ý$²»·ûºÏÌâÒâ
µ±m£¾0ʱ£¬Òª?t₁£¬t₂ʹµÃf£¨t₁£©=f£¨t₂£©£¬
ÄÇôÓÉÌâÒâÖªf£¨x£©µÄ¼«Öµµã±ØÔÚÇø¼ä£¨0£¬e£©ÄÚ£¬¼´$0£¼\frac{2}{m}£¼e$
µÃ$m£¾\frac{2}{e}$£¬ÇÒº¯Êýf£¨x£©ÔÚ$£¨{0£¬\frac{2}{m}}£©¡ý£¬£¨{\frac{2}{m}£¬e}£©¡ü$
ÓÉÌâÒâµÃg£¨x£©ÔÚ£¨0£¬e£©ÉϵÄÖµÓò°üº¬ÓÚf£¨x£©ÔÚ$£¨{0£¬\frac{2}{m}}£©ºÍ£¨{\frac{2}{m}£¬e}£©$ÉϵÄÖµÓò£¬
¡à$£¨{\frac{2}{m}£¬e}£©$ÄÚ£¬$\left\{{\begin{array}{l}{f£¨\frac{2}{m}£©¡Ü0}\\{f£¨e£©¡Ý1}\end{array}}\right.⇒m¡Ý\frac{3}{e-1}$£¬
ÏÂÃæÖ¤$t¡Ê£¨{0£¬\frac{2}{m}}]$ʱ£¬f£¨t£©¡Ý1£¬È¡t=e^-m£¬ÏÈÖ¤${e^{-m}}£¼\frac{2}{m}£¬¼´Ö¤2{e^m}-m£¾0$£®
Áîw£¨x£©=2e^x-x£¬¡à${w^'}£¨x£©=2{e^x}-1£¾0£¬ÔÚ[{\frac{3}{e-1}£¬+¡Þ}£©$ÄÚºã³ÉÁ¢£¬¡àw£¨x£©¡ü£¬¡à$w£¨x£©¡Ýw£¨\frac{3}{e-1}£©£¾0$£¬¡à2e^m-m£¾0£¬
ÔÙÖ¤f£¨e^-m£©¡Ý1£¬¡ß$f£¨{e^{-m}}£©=m{e^{-m}}+m£¾m¡Ý\frac{3}{e-1}£¾1$£¬
¡à$m¡Ý\frac{3}{e-1}$£®¡­£¨14·Ö£©

µãÆÀ ±¾Ì⿼²éº¯ÊýµÄµ¼ÊýµÄ×ÛºÏÓ¦Ó㬺¯ÊýµÄ¼«ÖµÒÔ¼°º¯ÊýµÄµ¥µ÷ÐÔµÄÅжÏÓëÓ¦Óã¬Ðº¯ÊýÒÔ¼°¹¹Ôì·¨µÄÓ¦Ó㬿¼²é×ۺϷÖÎöÎÊÌâ½â¾öÎÊÌâµÄÄÜÁ¦£®