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这题必须得用泰勒,我摊牌了(2022全国甲,12)

2022-12-20 20:48 作者:数学老顽童  | 我要投稿

(2022全国甲,12)已知a%3D%5Cfrac%7B31%7D%7B32%7Db%3D%5Ccos%20%5Cfrac%7B1%7D%7B4%7Dc%3D4%5Csin%20%5Cfrac%7B1%7D%7B4%7D,则(    )

A.c%3Eb%3Ea

B.b%3Ea%3Ec

C.a%3Eb%3Ec

D.a%3Ec%3Eb

解:令f%5Cleft(%20x%20%5Cright)%20%3D%5Ccos%20%20x%2B%5Cfrac%7B1%7D%7B2%7Dx%5E2-1

f'%5Cleft(%20x%20%5Cright)%20%3Dx-%5Csin%20%20x

f''%5Cleft(%20x%20%5Cright)%20%3D1-%5Ccos%20%20x%5Cgeqslant%200

所以f'%5Cleft(%20x%20%5Cright)%20%5Cnearrow%20

又因为f'%5Cleft(%200%20%5Cright)%20%3D0,所以:

x%5Cin%20%5Cleft(%20-%5Cinfty%20%2C0%20%5Cright)%20f'%5Cleft(%20x%20%5Cright)%20%3C0f%5Cleft(%20x%20%5Cright)%20%5Csearrow%20

x%5Cin%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20f'%5Cleft(%20x%20%5Cright)%20%3E0f%5Cleft(%20x%20%5Cright)%20%5Cnearrow%20

所以f%5Cleft(%20x%20%5Cright)%20%5Cgeqslant%20f%5Cleft(%20x%20%5Cright)%20_%7B%5Cmin%7D%3Df%5Cleft(%200%20%5Cright)%20%3D0

%5Ccolor%7Bred%7D%7B%5Ccos%20%20x%5Cgeqslant%201-%5Cfrac%7B1%7D%7B2%7Dx%5E2%7D,所以:

%5Ccolor%7Bred%7D%7Bb%7D%3D%5Ccos%20%5Cfrac%7B1%7D%7B4%7D%5Ccolor%7Bred%7D%7B%3E%7D1-%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20%5Cleft(%20%5Cfrac%7B1%7D%7B4%7D%20%5Cright)%20%5E2%3D%5Cfrac%7B31%7D%7B32%7D%3D%5Ccolor%7Bred%7D%7Ba%7D

g%5Cleft(%20x%20%5Cright)%20%3D%5Ctan%20%20x-xx%5Cin%20%5Cleft%5B%200%2C%5Cfrac%7B%5Cmathrm%7B%5Cpi%7D%7D%7B2%7D%20%5Cright)%20

g'%5Cleft(%20x%20%5Cright)%20%3D%5Cfrac%7B1%7D%7B%5Ccos%20%5E2x%7D-1%5Cgeqslant%200

所以g%5Cleft(%20x%20%5Cright)%20%5Cnearrow%20

所以g%5Cleft(%20x%20%5Cright)%20%5Cgeqslant%20g%5Cleft(%20x%20%5Cright)%20_%7B%5Cmin%7D%3Dg%5Cleft(%200%20%5Cright)%20%3D0

%5Ccolor%7Bred%7D%7Bx%5Cin%20%5Cleft%5B%200%2C%5Cfrac%7B%5Cmathrm%7B%5Cpi%7D%7D%7B2%7D%20%5Cright)%20%7D%5Ccolor%7Bred%7D%7B%5Ctan%20%20x%5Cgeqslant%20x%7D

%5Cfrac%7B%5Csin%20%20x%7D%7B%5Ccos%20%20x%7D%5Cgeqslant%20x

%5Ccolor%7Bred%7D%7B%5Cfrac%7B%5Csin%20%20x%7D%7Bx%7D%3E%20%5Ccos%20%20x%7D,所以

%5Ccolor%7Bred%7D%7Bc%7D%3D4%5Csin%20%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B%5Csin%20%5Cfrac%7B1%7D%7B4%7D%7D%7B%5Cfrac%7B1%7D%7B4%7D%7D%5Ccolor%7Bred%7D%7B%3E%7D%5Ccos%20%20%5Cfrac%7B1%7D%7B4%7D%3D%5Ccolor%7Bred%7D%7Bb%7D.

综上所述:c%3Eb%3Ea,选A.

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