Question

Let $f\left(x\right)=\left|\begin{array}{ccc}\sin x& 0& 2\cos x\\ 0& \sin x& 0\\ 2\cos x& 0& \sin x\end{array}\right|$ where $x\in (0,\pi ).$ Then total number of local maxima and local minima of $f\left(x\right)$ is

Answer 1

We have, $f\left(x\right)=\left|\begin{array}{ccc}\sin x& 0& 2\cos x\\ 0& \sin x& 0\\ 2\cos x& 0& \sin x\end{array}\right|$
$\Rightarrow f\left(x\right)={\sin }^{3}x-2\cos x\left(2\sin x\cos x\right)$
$\Rightarrow f\left(x\right)={\sin }^{3}x-4\sin x(1-{\sin }^{2}x)$
$\Rightarrow f\left(x\right)=5{\sin }^{3}x-4\sin x$
Differentiating w.r.t. $x,$ we get
$f^{\prime }\left(x\right)=15{\sin }^{2}x\cos x-4\cos x$
$\Rightarrow f^{\prime }\left(x\right)=15\cos x({\sin }^{2}x-\frac{4}{15})$
$\Rightarrow f^{\prime }\left(x\right)=15\cos x(\sin x-\frac{2}{\sqrt{15}})(\sin x+\frac{2}{\sqrt{15}})$
$\cos x=0$ at $x=\frac{\pi }{2}$
and $\sin x=\frac{2}{\sqrt{15}}$ at two points in $(0,\pi )$
$\therefore$ Total required points $=3$