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互为倒数的零点(2019课标Ⅱ(文)导数)

2022-09-27 21:32 作者:数学老顽童  | 我要投稿

(2019课标Ⅱ文,21)已知函数f%5Cleft(%20x%20%5Cright)%20%3D%5Cleft(%20x-1%20%5Cright)%20%5Cln%20%20x-x-1.证明:

(1)f%5Cleft(%20x%20%5Cright)%20存在唯一的极值点;

(2)f%5Cleft(%20x%20%5Cright)%20%3D0有且仅有两个实根,且两个实根互为倒数.

解:(1)求导,得

%5Cbegin%7Baligned%7D%0A%09f'%5Cleft(%20x%20%5Cright)%20%26%3D1%5Ccdot%20%5Cln%20x%2B%5Cleft(%20x-1%20%5Cright)%20%5Ccdot%20%5Cfrac%7B1%7D%7Bx%7D-1%5C%5C%0A%09%26%3D%5Cln%20x-%5Cfrac%7B1%7D%7Bx%7D%5C%5C%0A%5Cend%7Baligned%7D

易知%5Ccolor%7Bred%7D%7Bf'%5Cleft(%20x%20%5Cright)%20%5Cnearrow%20%7D

f'%5Cleft(%20%5Ccolor%7Bred%7D%7B1%7D%20%5Cright)%20%3D-1%5Ccolor%7Bred%7D%7B%3C0%7D

f'%5Cleft(%20%5Ccolor%7Bred%7D%7B%5Cmathrm%7Be%7D%7D%20%5Cright)%20%3D1-%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%7D%5Ccolor%7Bred%7D%7B%3E0%7D

故存在唯一%5Ccolor%7Bred%7D%7Bx_0%7D%5Cin%20%5Cleft(%201%2C%5Cmathrm%7Be%7D%20%5Cright)%20,使得f'%5Cleft(%20x_0%20%5Cright)%20%3D0.

x%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft(%200%2Cx_0%20%5Cright)%7D%20,f'%5Cleft(%20x%20%5Cright)%20%3C0,f%5Cleft(%20x%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Csearrow%20%7D;

x%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft(%20x_0%2C%2B%5Cinfty%20%5Cright)%20%7D,f'%5Cleft(%20x%20%5Cright)%20%3E0,f%5Cleft(%20x%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Cnearrow%7D;

f%5Cleft(%20x%20%5Cright)%20存在唯一的极小值点x_0,无极大值点,证毕.

(2)由(1)可知%5Cln%20x_0-%5Cfrac%7B1%7D%7Bx_0%7D%3D0

%5Cln%20x_0%3D%5Cfrac%7B1%7D%7Bx_0%7D,所以(隐零点代换

%5Cbegin%7Baligned%7D%0A%09f%5Cleft(%20x%20%5Cright)%20_%7B%5Cmin%7D%26%3Df%5Cleft(%20x_0%20%5Cright)%5C%5C%0A%09%26%3D%5Cleft(%20x_0-1%20%5Cright)%5Ccolor%7Bred%7D%7B%20%5Cln%20x_0%7D-x_0-1%5C%5C%0A%09%26%3D%5Cleft(%20x_0-1%20%5Cright)%20%5Ccdot%20%5Ccolor%7Bred%7D%7B%5Cfrac%7B1%7D%7Bx_0%7D%7D-x_0-1%5C%5C%0A%09%26%3D-%5Cleft(%20x_0%2B%5Cfrac%7B1%7D%7Bx_0%7D%20%5Cright)%20%5Ccolor%7Bred%7D%7B%3C0%7D%5C%5C%0A%5Cend%7Baligned%7D

又因为

f%5Cleft(%20%5Ccolor%7Bred%7D%7B%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%5E2%7D%7D%20%5Cright)%20%3D1-%5Cfrac%7B3%7D%7B%5Cmathrm%7Be%7D%5E2%7D%5Ccolor%7Bred%7D%7B%3E0%7D

f%5Cleft(%20%5Ccolor%7Bred%7D%7B%5Cmathrm%7Be%7D%5E2%7D%20%5Cright)%20%3D%5Cmathrm%7Be%7D%5E2-3%5Ccolor%7Bred%7D%7B%3E0%7D

故存在%5Ccolor%7Bred%7D%7Bx_1%7D%5Cin%20%5Cleft(%20%5Cfrac%7B1%7D%7B%5Cmathrm%7Be%7D%5E2%7D%2Cx_0%20%5Cright)%20%5Ccolor%7Bred%7D%7Bx_2%7D%5Cin%20%5Cleft(%20x_0%2C%5Cmathrm%7Be%7D%5E2%20%5Cright)%20

使得f%5Cleft(%20x_1%20%5Cright)%20%3Df%5Cleft(%20x_2%20%5Cright)%20%3D0.(横屏观看

又因为

%5Cbegin%7Baligned%7D%0A%09f%5Cleft(%20%5Cfrac%7B1%7D%7Bx_1%7D%20%5Cright)%20%26%3D%5Cleft(%20%5Cfrac%7B1%7D%7Bx_1%7D-1%20%5Cright)%20%5Cln%20%5Cfrac%7B1%7D%7Bx_1%7D-%5Cfrac%7B1%7D%7Bx_1%7D-1%5C%5C%0A%09%26%3D%5Cfrac%7B%5Cleft(%20x_1-1%20%5Cright)%20%5Cln%20x_1-x_1-1%7D%7Bx_1%7D%5C%5C%0A%09%26%3D%5Cfrac%7B%5Ccolor%7Bred%7D%7Bf%5Cleft(%20x_1%20%5Cright)%7D%7D%7Bx_1%7D%5C%5C%0A%09%26%3D%5Cfrac%7B0%7D%7Bx_1%7D%3D0%5C%5C%0A%5Cend%7Baligned%7D

%5Cfrac%7B1%7D%7Bx_1%7D%5Cneq%20x_1

所以%5Ccolor%7Bred%7D%7B%5Cfrac%7B1%7D%7Bx_1%7D%3Dx_2%7D,证毕.

留个思考题:

为什么会出现两个互为倒数的零点?它是如何编出来的?

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