sin(x²/2) 성질 보기

ㄱ. $0 < x ^{2} < 1$ 이므로, $\displaystyle x ^{2} \sin \frac{x ^{2}}{2} < \sin \frac{x ^{2}}{2}$ $\cdots \cdots \text{①}$ $\displaystyle 0 < \frac{x ^{2}}{2} < \frac{1}{2} < \frac{\pi}{6}$ 이므로, $\displaystyle \sin \frac{x ^{2}}{2} < \cos \frac{x ^{2}}{2}$ $\cdots \cdots \text{ ②}$ ①, ② 에 의해 $\displaystyle x ^{2} \sin \frac{x ^{2}}{2} < f ( x ) < \cos \frac{x ^{2}}{2}$ (참) ㄴ. $\displaystyle f' ( x ) = x \cos \frac{x ^{2}}{2}$ , $\displaystyle f'' ( x ) = \cos \frac{x ^{2}}{2} - x ^{2} \sin \frac{x ^{2}}{2} > 0$ ($\because$ㄱ. 에 의해 ) 따라서, $y = f ( x )$ 는 $0 < x < 1$ 에서 아래로 볼록. (거짓) ㄷ. $\displaystyle \sin \frac{1}{2}$ $\displaystyle y = \left( \sin \frac{1}{2} \right) x$ $y = f ( x )$

$0 < x < 1$ 에서 아래로 볼록이므로, [그림1]에서 $\displaystyle f ( x ) \leq \left( \sin \frac{1}{2} \right) x$ 따라서, $\displaystyle \int _{0} ^{1} {f ( x )} dx \leq \int _{0} ^{1} {\left( \sin \frac{1}{2} \right) x} dx = \frac{1}{2} \mathrm{\sin} \frac{1}{2}$ (참)