Let f(x)=4 ^3x+3√3 ^2x-10.

Question

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

Accepted Answer

$$\begin{aligned}
& f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10 \
& f^{\prime}(x)=12 \cos ^2 x \cdot(-\sin x)+6 \sqrt{3} \cos x \cdot(-\sin x)=0 \
& =-6 \sqrt{3} \cos x \cdot \sin x\left(1+\frac{2}{\sqrt{3}} \cos x\right)=0 \
& \cos x=0, \sin x=0, \cos x=\frac{-\sqrt{3}}{2}
\end{aligned}$$

Sign of $f(x)$

$$\therefore \text { Maxima at } \frac{5 \pi}{6}, \frac{7 \pi}{6}$$

Solutions · 1