Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. For $0 \lt t \lt \pi$, let $R_t$ be the region bounded by the curve $y=f(x)$, the line $x=0$, the line $y=0$ and the line $x=t$. Suppose the area of $R_t$ is $\frac{t^2}{2}(1-\cos t)$ for $0 \lt t \lt \pi$. Then

• A

  $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4}\left(1+\frac{\pi}{4}\right)$
• Correct answer
  B

  $f\left(\frac{\pi}{2}\right)=\frac{\pi}{2}\left(1+\frac{\pi}{4}\right)$
• C

  $f\left(\frac{\pi}{2}\right)=\pi\left(1+\frac{\pi}{8}\right)$
• D

  $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4 \sqrt{2}}\left(1-\frac{\pi}{8}\right)$

Hence, the correct answer is option: 'B'.