Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in interval $(0,2 \pi)$ is

• A

  3
• B

  4
• C

  1
• Correct answer
  D

  2

Solution

$\begin{aligned} & \mathrm{f}(\mathrm{x})=4 \cos ^3(\mathrm{x})+3 \sqrt{3} \cos ^2(\mathrm{x})-10 ; \mathrm{x} \in(0,2 \pi) \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=12 \cos ^2 \mathrm{x}[-\sin (\mathrm{x})]+3 \sqrt{3}(2 \cos (\mathrm{x}))[-\sin (\mathrm{x})] \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-6 \sin (\mathrm{x}) \cos (\mathrm{x})[2 \cos (\mathrm{x})+\sqrt{3}]\end{aligned}$
local maxima at $\mathrm{x}=\frac{5 \pi}{6}, \frac{7 \pi}{6}$

Hence, the correct answer is option: 'D'.