Question

For $x\in (0,\frac{5\pi }{4})$ , define $f\left(x\right)={\int }_0^x\sqrt{t}$ sint dt. Then $f\left(x\right)$ has

• A. local maximum at $\pi$ and $2\pi$
• B. local minimum at $\pi$ and $2\pi$
• C. local minimum at $\pi$ and local maximum at $2\pi$
• D. local maximum at $\pi$ and local minimum at $2\pi$

Answer 1

The correct option is D local maximum at $\pi$ and local minimum at $2\pi$

$f\left(x\right)={\int }_0^x\sqrt{t}$ $\sin t$
$f^{\prime }\left(x\right)=\sqrt{x}\sin x$
Given $xϵ(0,\frac{5\pi }{4})$
$f^{\prime }\left(x\right)$ changes sign from +ve to -ve at $\pi$
$f^{\prime }\left(x\right)$ changes sign from -ve to +ve at $2\pi$
$f\left(x\right)$ has local maximum at $\pi$ and local minima at $2\pi$
Hence, option 'D' is correct.