Question

The function $f\left(x\right)=3co{s}^{4}x+10co{s}^{3}x+6co{s}^{2}x-3,\left(0\le x\le \pi \right)$ is $-$

• A. Increasing in $\left(\frac{\pi }{2},\frac{2\pi }{3}\right)$
• B. Increasing in $\left(0,\frac{\pi }{2}\right)\cup \left(\frac{2\pi }{3},\pi \right)$
• C. Decreasing in $\left(\frac{\pi }{2},\frac{2\pi }{3}\right)$
• D. All of above

Answer 1

Answer: (A)

$f^{\prime }(x)=-12co{s}^{3}xsinx-30co{s}^{2}xsinx-12cosxsin$
$x=-6sinxcosx(cosx+2)(2cosx+1)$
$f^{\prime }(x)=0$, for $x=0,\frac{\pi }{2},\frac{2\pi }{3},\pi$
Clearly, $f^{\prime }\left(x\right)>0$ for $\frac{\pi }{2}<x<\frac{2\pi }{3}$
And $f^{\prime }\left(x\right)<0$ ; for $0<x<\frac{\pi }{2}$ or $\frac{2\pi }{3}<x<\pi$