Question

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

$f ' (x) = - 12\, cos^3\, x sin \,x - 30 cos^2x\, sin\, x - 12\, cos\, x \,sin$
$x = - 6 \,sin\, x\, cos\,x (cos\, x + 2) (2 cos \,x + 1)$
$f ' (x) = 0$, for $x=0, \frac{\pi}{2}, \frac{2\pi}{3}, \pi$
Clearly, $f ' \left(x\right) > 0$ for $\frac{\pi}{2} < x < \frac{2\pi}{3}$
And $f ' \left(x\right) < 0$ ; for $0 < x < \frac{\pi}{2}$ or $\frac{2\pi}{3} < x < \pi$