已知 $a=\frac{31}{32}, b=\cos \frac{1}{4}, c=4 \sin \frac{1}{4}$, 则
$ \text{A.} $ $c > b > a$ $ \text{B.} $ $b > a > c$ $ \text{C.} $ $a > b > c$ $ \text{D.} $ $a > c > b$
【答案】 A

【解析】 $f(x)=\sin x-x \cos x, g(x)=\cos x-\left(1-\frac{1}{2} x^2\right), 0 < x < 1$.

$$
\begin{aligned}
& f^{\prime}(x)=x \sin x > 0, g^{\prime}(x)=-\sin x+x > 0 . \\
& \Rightarrow f\left(\frac{1}{4}\right) > f(0)=0, g\left(\frac{1}{4}\right) > g(0)=0 . \Rightarrow \sin \frac{1}{4}-\frac{1}{4} \cos \frac{1}{4} > 0, \cos \frac{1}{4} > 1-\frac{1}{2}\left(\frac{1}{4}\right)^2=\frac{31}{32} .
\end{aligned}
$$
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