Question

If $f(x)=\sin \left(\frac{\pi}{3}[x]-x^{2}\right)$, where $2< x< 3$ and $[.]$ represents greatest integer function, then $f'\left(\sqrt{\frac{5 \pi}{3}}\right)$ is equal to

• A. $\sqrt{\frac{5 \pi}{3}}$
• B. $2 \sqrt{\frac{5 \pi}{3}}$
• C. $-2 \sqrt{\frac{5 \pi}{3}}$
• D. $-\sqrt{\frac{5 \pi}{3}}$

Answer 1

Answer: (B)

Since, $2<\,x<\,3, $
$\therefore [x]=2$
$\therefore f(x)=\sin \left(\frac{2 \pi}{3}-x^{2}\right)$
$\therefore f'(x)=\cos \left(\frac{2 \pi}{3}-x^{2}\right)(-2 x) $
$\therefore f'\left(\sqrt{\frac{5 \pi}{3}}\right)=2 \sqrt{\frac{5 \pi}{3}}$