Question

Let $f\left(x\right)=\sin x+2{\cos }^{2}x,\frac{\pi }{4}\le x\le \frac{3\pi }{4}$. Then f(x) attains its:

• A. minimum at $x=\frac{\pi }{4}$
• B. maximum at $x=\frac{\pi }{2}$
• C. minimum at $x=\frac{\pi }{2}$
• D. maximum at $x=si{n}^{-1}\left(\frac{1}{4}\right)$

Answer 1

The correct option is C minimum at $x=\frac{\pi }{2}$

$\Rightarrow f^{{}^{\prime }}\left(x\right)=\cos x+4\cos x(-\sin x)$

$\Rightarrow f^{{}^{\prime }}\left(x\right)=0$ for the minima or maxima to occur

$\Rightarrow \cos x(1-4\sin x)=0$

$\Rightarrow x=\frac{\pi }{2}$ or $x={\sin }^{-1}\left(\frac{1}{4}\right)$

$\Rightarrow f^{{}^{\prime \prime }}\left(x\right)=\cos x(-4\sin x)-(1-4\sin x)\sin x$

If $f^{{}^{\prime \prime }}\left(x\right)>0$, minimum occurs at x

If $f^{{}^{\prime \prime }}\left(x\right)<0$, maximum occurs at x

$\Rightarrow f^{{}^{\prime \prime }}\left(\frac{\pi }{2}\right)=3>0$

$\therefore$ at $x=\frac{\pi }{2}$, minimum occurs