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利用第一問的結(jié)論(2020課標(biāo)Ⅱ(文)導(dǎo)數(shù))

2022-09-26 19:59 作者:數(shù)學(xué)老頑童  | 我要投稿

(2020課標(biāo)Ⅱ文,21)已知函數(shù)f%5Cleft(%20x%20%5Cright)%20%3D2%5Cln%20%20x%2B1.

(1)若f%5Cleft(%20x%20%5Cright)%20%5Cleqslant%202x%2Bc,求c的取值范圍;

(2)設(shè)a%3E0,討論函數(shù)g%5Cleft(%20x%20%5Cright)%20%3D%5Cfrac%7Bf%5Cleft(%20x%20%5Cright)%20-f%5Cleft(%20a%20%5Cright)%7D%7Bx-a%7D的單調(diào)性.

解:(1)f%5Cleft(%20x%20%5Cright)%20%5Cleqslant%202x%2Bc

2%5Cln%20%20x%2B1%5Cleqslant%202x%2Bc,

c%5Cgeqslant%202%5Cln%20%20x-2x%2B1,

h%5Cleft(%20x%20%5Cright)%20%3D2%5Cln%20%20x-2x%2B1

h'%5Cleft(%20x%20%5Cright)%20%3D%5Cfrac%7B2%7D%7Bx%7D-2%3D%5Cfrac%7B2%5Cleft(%201-x%20%5Cright)%7D%7Bx%7D,

h'%5Cleft(%20x%20%5Cright)%20%3D0,得x%3D1

當(dāng)x%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft(%200%2C1%20%5Cright)%20%7D,h'%5Cleft(%20x%20%5Cright)%20%3E0,h%5Cleft(%20x%20%5Cright)%20%20%5Ccolor%7Bred%7D%7B%5Cnearrow%20%7D,

當(dāng)x%5Cin%20%5Ccolor%7Bred%7D%7B%20%5Cleft(%201%2C%2B%5Cinfty%20%5Cright)%20%7D,h'%5Cleft(%20x%20%5Cright)%20%3C0,h%5Cleft(%20x%20%5Cright)%20%20%5Ccolor%7Bred%7D%7B%5Csearrow%20%7D,

%20%5Ccolor%7Bred%7D%7Bh%5Cleft(%20x%20%5Cright)%20_%7B%5Cmax%7D%7D%3Dh%5Cleft(%201%20%5Cright)%20%3D%20%5Ccolor%7Bred%7D%7B-1%7D,

%20%5Ccolor%7Bred%7D%7Bc%5Cin%20%5Cleft%5B%20-1%2C%2B%5Cinfty%20%5Cright)%20%7D.

(2)

%5Cbegin%7Baligned%7D%0A%09%20%5Ccolor%7Bred%7D%7Bg%5Cleft(%20x%20%5Cright)%7D%20%26%3D%5Cfrac%7B2%5Cln%20x%2B1-%5Cleft(%202%5Cln%20a%2B1%20%5Cright)%7D%7Bx-a%7D%5C%5C%0A%09%26%3D%20%5Ccolor%7Bred%7D%7B%5Cfrac%7B2%5Cln%20x-2%5Cln%20a%7D%7Bx-a%7D%7D%5C%5C%0A%5Cend%7Baligned%7D

其中x%5Cin%20%20%5Ccolor%7Bred%7D%7B%5Cleft(%200%2Ca%20%5Cright)%20%5Ccup%20%5Cleft(%20a%2C%2B%5Cinfty%20%5Cright)%20%7D,

%5Cbegin%7Baligned%7D%0A%09g'%5Cleft(%20x%20%5Cright)%20%26%3D%5Cfrac%7B%5Cfrac%7B2%7D%7Bx%7D%5Cleft(%20x-a%20%5Cright)%20-%5Cleft(%202%5Cln%20x-2%5Cln%20a%20%5Cright)%20%5Ccdot%201%7D%7B%5Cleft(%20x-a%20%5Cright)%20%5E2%7D%5C%5C%0A%09%26%3D%5Cfrac%7B2%7D%7B%5Cleft(%20x-a%20%5Cright)%20%5E2%7D%5Ccdot%20%5Cleft(%20%20%5Ccolor%7Bred%7D%7B%5Cln%20%5Cfrac%7Ba%7D%7Bx%7D-%5Cfrac%7Ba%7D%7Bx%7D%2B1%7D%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D

由(1)可知2%5Cln%20%20x-2x%2B1%5Cleqslant%20-1

%5Ccolor%7Bred%7D%7B%5Cforall%20x%5Cin%20%5Cleft(0%2C%5Cinfty%5Cright)%7D,%20%5Ccolor%7Bred%7D%7B%5Cln%20%20x-x%2B1%5Cleqslant%200%7D

所以%20%5Ccolor%7Bred%7D%7B%5Cln%20%5Cfrac%7Ba%7D%7Bx%7D-%5Cfrac%7Ba%7D%7Bx%7D%2B1%5Cleqslant%200%7D,

所以g'%5Cleft(%20x%20%5Cright)%20%5Cleqslant%200,

所以g%5Cleft(%20x%20%5Cright)%20%5Cleft(%200%2Ca%20%5Cright)%20%5Cleft(%20a%2C%2B%5Cinfty%20%5Cright)%20%20%5Ccolor%7Bred%7D%7B%5Csearrow%20%7D.

命題背景:

%5Cbbox%5B%23def%2C5px%2Cborder%3A1px%20solid%5D%7B%0A%5Cforall%20x%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft(0%2C%5Cinfty%5Cright)%7D%2C%5Cln%20%20x%5Ccolor%7Bred%7D%7B%5Cleqslant%7D%20x-1.%7D


利用第一問的結(jié)論(2020課標(biāo)Ⅱ(文)導(dǎo)數(shù))的評(píng)論 (共 條)

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