Question

The interval in which the function $f\left(x\right)={\sin }^{2}x,x\in (0,\pi )$ is concave down

• A. $(\frac{\pi }{2},\pi )$
• B. $(0,\frac{\pi }{4})$
• C. $(\frac{\pi }{4},\frac{3\pi }{4})$
• D. $(\frac{3\pi }{4},\pi )$

Answer 1

The correct option is C $(\frac{\pi }{4},\frac{3\pi }{4})$

Given, $f\left(x\right)={\sin }^{2}x,x\in (0,\pi )f^{\prime }\left(x\right)=\sin 2x{f}^{\prime \prime }\left(x\right)=2\cos 2x$
For concave down$,f^{\prime \prime }\left(x\right)<0$
$\Rightarrow \cos 2x<0,x\in (0,\pi )$
$\Rightarrow x\in (\frac{\pi }{4},\frac{3\pi }{4})$