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构造函数证明不等式(2023新高考Ⅰ导数)

2023-07-10 11:03 作者:数学老顽童  | 我要投稿

(2023新高考Ⅰ,19)已知函数f%5Cleft(%20x%20%5Cright)%20%3Da%5Cleft(%20%5Cmathrm%7Be%7D%5Ex%2Ba%20%5Cright)%20-x.

(1)讨论f%5Cleft(%20x%20%5Cright)%20的单调性;

(2)证明:当a%3E0时,f%5Cleft(%20x%20%5Cright)%20%3E2%5Cln%20%20a%2B%5Cfrac%7B3%7D%7B2%7D.

解:(1)f'%5Cleft(%20x%20%5Cright)%20%3Da%5Cmathrm%7Be%7D%5Ex-1

a%5Cleqslant%200

f'%5Cleft(%20x%20%5Cright)%20%3C0f%5Cleft(%20x%20%5Cright)%20%5Csearrow%20

a%3E0

f'%5Cleft(%20x%20%5Cright)%20%3D0,得x%3D%5Ccolor%7Bred%7D%7B%5Cln%20%5Cfrac%7B1%7D%7Ba%7D%7D

x%5Cin%20%5Ccolor%7Bred%7D%7B%5Cleft(%20-%5Cinfty%20%2C%5Cln%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cright)%20%7Df'%5Cleft(%20x%20%5Cright)%20%3C0f%5Cleft(%20x%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Csearrow%20%7D

x%5Cin%5Ccolor%7Bred%7D%7B%20%5Cleft(%20%5Cln%20%5Cfrac%7B1%7D%7Ba%7D%2C%2B%5Cinfty%20%5Cright)%20%7Df'%5Cleft(%20x%20%5Cright)%20%3E0f%5Cleft(%20x%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Cnearrow%20%7D

(2)由(1)可知,当a%3E0时,

f%5Cleft(%20x%20%5Cright)%20%5Cgeqslant%20f%5Cleft(%20%5Cln%20%5Cfrac%7B1%7D%7Ba%7D%20%5Cright)%20%3D%5Ccolor%7Bred%7D%7B1%2Ba%5E2%2B%5Cln%20%20a%7D

故只需证:1%2Ba%5E2%2B%5Cln%20%20a%3E2%5Cln%20%20a%2B%5Cfrac%7B3%7D%7B2%7D

即证:%5Ccolor%7Bred%7D%7Ba%5E2-%5Cln%20%20a-%5Cfrac%7B1%7D%7B2%7D%3E0%7D.

g%5Cleft(%20a%20%5Cright)%20%3Da%5E2-%5Cln%20%20a-%5Cfrac%7B1%7D%7B2%7D,则

g'%5Cleft(%20a%20%5Cright)%20%3D2a-%5Cfrac%7B1%7D%7Ba%7D%3D%5Cfrac%7B%5Cleft(%20%5Csqrt%7B2%7Da%2B1%20%5Cright)%20%5Ccolor%7Bred%7D%7B%5Cleft(%20%5Csqrt%7B2%7Da-1%20%5Cright)%7D%7D%7Ba%7D

g'%5Cleft(%20a%20%5Cright)%20%3D0,得a%3D%5Ccolor%7Bred%7D%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%7D

a%5Cin%20%5Cleft(%200%2C%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cright)%20g'%5Cleft(%20a%20%5Cright)%20%3C0g%5Cleft(%20a%20%5Cright)%20%5Csearrow%20

a%5Cin%20%5Cleft(%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%2C%2B%5Cinfty%20%5Cright)%20g'%5Cleft(%20a%20%5Cright)%20%3E0g%5Cleft(%20a%20%5Cright)%20%5Cnearrow%20

%5Ccolor%7Bred%7D%7Bg%5Cleft(%20a%20%5Cright)%20%5Cgeqslant%20g%5Cleft(%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cright)%20%3D%5Cfrac%7B%5Cln%20%202%7D%7B2%7D%3E0%7D,证毕.

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