專為“同構(gòu)”這碟醋包的餃子（2022新高考Ⅰ導(dǎo)數(shù)）

2022-10-06 23:32 作者:數(shù)學(xué)老頑童 0人讀過(guò) | 我要投稿

解：（1）若 $a%5Cleqslant%200$ ，則 $f%5Cleft(%20x%20%5Cright)%20$ 與 $g%5Cleft(%20x%20%5Cright)%20$ 皆 $%5Csearrow%20$ ，

無(wú)最小值，不合題意.

當(dāng) $a%3E0$ 時(shí)，

$f'%5Cleft(%20x%20%5Cright)%20%3D%5Cmathrm%7Be%7D%5Ex-a$ ，

令 $f'%5Cleft(%20x%20%5Cright)%20%3D0$ ，得 $x%3D%5Cln%20%20a$ ，

其單調(diào)性如下表所示：

$%5Cbegin%7Barray%7D%7Bc%7Cc%7Cc%7D%0A%09x%26%09%09%5Ccolor%7Bred%7D%7B%5Cleft(%20-%5Cinfty%20%2C%5Cln%20a%20%5Cright)%7D%26%09%09%5Ccolor%7Bred%7D%7B%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%7D%5C%5C%0A%09%5Chline%0A%09f%5Cprime%5Cleft(%20x%20%5Cright)%26%09%09-%26%09%09%2B%5C%5C%0A%09%5Chline%0A%09f%5Cleft(%20x%20%5Cright)%26%09%09%5Ccolor%7Bred%7D%7B%5Csearrow%7D%26%09%09%5Ccolor%7Bred%7D%7B%5Cnearrow%7D%5C%5C%0A%5Cend%7Barray%7D$

故 $%5Ccolor%7Bred%7D%7Bf%5Cleft(%20x%20%5Cright)%20_%7B%5Cmin%7D%7D%3Df%5Cleft(%20%5Cln%20%20a%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7Ba-a%5Cln%20%20a%7D$ .

$g'%5Cleft(%20x%20%5Cright)%20%3Da-%5Cfrac%7B1%7D%7Bx%7D$ ，

令 $g'%5Cleft(%20x%20%5Cright)%20%3D0$ ，得 $x%3D%5Cfrac%7B1%7D%7Ba%7D$ ；

其單調(diào)性如下表所示：

$%5Cbegin%7Barray%7D%7Bc%7Cc%7Cc%7D%0A%09x%26%09%09%5Ccolor%7Bred%7D%7B%5Cleft(%200%2C%5Cfrac%7B1%7D%7Ba%7D%20%5Cright)%7D%26%09%09%5Ccolor%7Bred%7D%7B%5Cleft(%20%5Cfrac%7B1%7D%7Ba%7D%2C%2B%5Cinfty%20%5Cright)%7D%5C%5C%0A%09%5Chline%0A%09g%5Cprime%5Cleft(%20x%20%5Cright)%26%09%09-%26%09%09%2B%5C%5C%0A%09%5Chline%0A%09g%5Cleft(%20x%20%5Cright)%26%09%09%5Ccolor%7Bred%7D%7B%5Csearrow%7D%26%09%09%5Ccolor%7Bred%7D%7B%5Cnearrow%7D%5C%5C%0A%5Cend%7Barray%7D$

故 $%5Ccolor%7Bred%7D%7Bg%5Cleft(%20x%20%5Cright)%20_%7B%5Cmin%7D%7D%3Dg%5Cleft(%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7B1%2B%5Cln%20%20a%7D$ .

依題意： $a-a%5Cln%20%20a%3D1%2B%5Cln%20%20a$ ，

即 $%5Cln%20%20a%2B%5Cfrac%7B2%7D%7Ba%2B1%7D-1%3D0$ .

令 $h%5Cleft(%20a%20%5Cright)%20%3D%5Cln%20%20a%2B%5Cfrac%7B2%7D%7Ba%2B1%7D-1$ ，則

$%5Ccolor%7Bred%7D%7Bh'%5Cleft(%20a%20%5Cright)%7D%20%3D%5Cfrac%7B1%7D%7Ba%7D-%5Cfrac%7B2%7D%7B%5Cleft(%20a%2B1%20%5Cright)%20%5E2%7D%3D%5Cfrac%7Ba%5E2%2B1%7D%7Ba%5Cleft(%20a%2B1%20%5Cright)%20%5E2%7D%5Ccolor%7Bred%7D%7B%3E0%7D$

故 $h%5Cleft(%20a%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Cnearrow%20%7D$ ，

故 $h%5Cleft(%20a%20%5Cright)%20$ 最多僅有一個(gè)零點(diǎn)，

又因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=h%5Cleft(%201%20%5Cright)%20%3D0" alt="h%5Cleft(%201%20%5Cright)%20%3D0">，故 $%5Ccolor%7Bred%7D%7Ba%3D1%7D$ .

（2）由（1）可知

$f%5Cleft(%20x%20%5Cright)%20%3D%5Cmathrm%7Be%7D%5Ex-x$ ， $g%5Cleft(%20x%20%5Cright)%20%3Dx-%5Cln%20%20x$ ，

$%5Cmathrm%7Be%7D%5Ex-x%5Cgeqslant%201$ ， $x-%5Cln%20%20x%5Cgeqslant%201$ ，

所以 $%7B%5Ccolor%7Bred%7D%20%7B%5Cln%20%20x%3Cx%3C%5Cmathrm%7Be%7D%5Ex%7D%7D$ ，

并且 $%7B%5Ccolor%7Bred%7D%20%7Bf%5Cleft(%20%5Cln%20%20x%20%5Cright)%20%3Dg%5Cleft(%20x%20%5Cright)%20%7D%7D$ ， $%7B%5Ccolor%7Bred%7D%20%7Bg%5Cleft(%20%5Cmathrm%7Be%7D%5Ex%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20%7D%7D$

令 $h%5Cleft(%20x%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20-g%5Cleft(%20x%20%5Cright)%20$ ， $x%5Cin%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20$

當(dāng) $x%5Cin%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20$ 時(shí)， $%5Cln%20%20x%5Cin%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20$ ，

則 $x$ 與 $%5Cln%20%20x$ 同屬于區(qū)間 $%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20$ ，

因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=f%5Cleft(%20x%20%5Cright)%20" alt="f%5Cleft(%20x%20%5Cright)%20">在該區(qū)間 $%5Cnearrow%20$ ，所以：

$h%5Cleft(%20x%20%5Cright)%20%3Df%5Cleft(%20x%20%5Cright)%20-%7B%5Ccolor%7Bred%7D%20%7Bg%5Cleft(%20x%20%5Cright)%20%7D%7D%3Df%5Cleft(%20x%20%5Cright)%20-%7B%5Ccolor%7Bred%7D%7B%20f%5Cleft(%20%5Cln%20%20x%20%5Cright)%7D%20%7D%3E0$

當(dāng) $x%5Cin%20%5Cleft(%200%2C1%20%5Cright)%20$ ，易知 $h%5Cleft(%20x%20%5Cright)%20%5Cnearrow%20$ ；

又因?yàn)?/p>

$%5Cbegin%7Baligned%7D%0A%09h%5Cleft(%20%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%7D%20%5Cright)%20%26%3D%5Cmathrm%7Be%7D%5E%7B%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%7D%7D-%5Cleft(%201%2B%5Cfrac%7B2%7D%7B%5Cmathrm%7Be%7D%7D%20%5Cright)%5C%5C%0A%09%26%3C%5Cmathrm%7Be%7D%5E%7B%5Cfrac%7B1%7D%7B%5Ccolor%7Bred%7D%7B2%7D%7D%7D-%5Cleft(%201%2B%5Cfrac%7B2%7D%7B%5Ccolor%7Bred%7D%7B3%7D%7D%20%5Cright)%5C%5C%0A%09%26%3D%5Csqrt%7B%5Cmathrm%7Be%7D%7D-%5Cfrac%7B5%7D%7B3%7D%3C0%5C%5C%0A%5Cend%7Baligned%7D$

$h%5Cleft(%201%20%5Cright)%20%3D%5Cmathrm%7Be%7D-2%3E0$ ，

故存在唯一 $x_0%5Cin%20%5Cleft(0%2C%2B%5Cpropto%20%20%5Cright)%20$ ，使 $h%5Cleft(%20x_0%20%5Cright)%20%3D0$ ，

即 $%5Ccolor%7Bred%7D%7Bf%5Cleft(%20x_0%20%5Cright)%20%3Dg%5Cleft(%20x_0%20%5Cright)%20%7D$ ，

即 $%5Cbbox%5B%23def%2C5px%2Cborder%3A0px%20solid%5D%7B%0A%5Cmathrm%7Be%7D%5E%7Bx_0%7D%2B%5Cln%20%20x_0%3D2x_0%7D$ .

又因?yàn)椋?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%7B%5Ccolor%7Bred%7D%20%7Bf%5Cleft(%20%5Cln%20%20x_0%20%5Cright)%20%3Dg%5Cleft(%20x_0%20%5Cright)%20%7D%7D" alt="%7B%5Ccolor%7Bred%7D%20%7Bf%5Cleft(%20%5Cln%20%20x_0%20%5Cright)%20%3Dg%5Cleft(%20x_0%20%5Cright)%20%7D%7D">，

$%7B%5Ccolor%7Bred%7D%20%7Bg%5Cleft(%20%5Cmathrm%7Be%7D%5E%7Bx_0%7D%20%5Cright)%20%3Df%5Cleft(%20x_0%20%5Cright)%20%7D%7D$ ,

所以： $%5Ccolor%7Bred%7D%20%7Bf%5Cleft(%20%5Cln%20%20x_0%20%5Cright)%20%3Df%5Cleft(%20x_0%20%5Cright)%20%7D$ ，

$%5Ccolor%7Bred%7D%20%7Bg%5Cleft(%20%5Cmathrm%7Be%7D%5E%7Bx_0%7D%20%5Cright)%20%3Dg%5Cleft(%20x_0%20%5Cright)%20%7D$ .

因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Cln%20%20x_0%5Cne%20x_0%5Cne%20%5Cmathrm%7Be%7D%5E%7Bx_0%7D" alt="%5Cln%20%20x_0%5Cne%20x_0%5Cne%20%5Cmathrm%7Be%7D%5E%7Bx_0%7D">

故存在直線 $y%3Db$ ，其與兩條曲線 $y%3Df%5Cleft(%20x%20%5Cright)%20$ 和 $y%3Dg%5Cleft(%20x%20%5Cright)%20$ 共有三個(gè)不同的交點(diǎn)： $%5Cleft(%20%7B%5Ccolor%7Bred%7D%20%7B%5Cln%20%20x_0%7D%7D%2Cb%20%5Cright)%20$ 、 $%5Cleft(%20%7B%5Ccolor%7Bred%7D%7B%20x_0%7D%7D%2Cb%20%5Cright)%20$ 、 $%5Cleft(%20%7B%5Ccolor%7Bred%7D%20%7B%5Cmathrm%7Be%7D%5E%7Bx_0%7D%7D%7D%2Cb%20%5Cright)$ ，且 $%5Cbbox%5B%23def%2C5px%2Cborder%3A0px%20solid%5D%7B%0A%5Cmathrm%7Be%7D%5E%7Bx_0%7D%2B%5Cln%20%20x_0%3D2x_0%7D$ .