Let f(x) = 4 cos^3 x + 3√3 cos^2 x - 10. The number of points of local maxima of f in interval (0, 2π) is:

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, 2π) is:

(1) 1 

(2) 2 

(3) 3 

(4) 4

Answer 1

Correct option is (2) 2   

\(f(x)=4 \cos ^{3}(x)+3 \sqrt{3} \cos ^{2}(x)-10 ; x \in(0,2 \pi)\)

\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=12 \cos ^{2} \mathrm{x}[-\sin (\mathrm{x})]+3 \sqrt{3}(2 \cos (\mathrm{x}))[-\sin (\mathrm{x})]\)

\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-6 \sin (\mathrm{x}) \cos (\mathrm{x})[2 \cos (\mathrm{x})+\sqrt{3}]\)

local maxima \(\mathrm{x}=\frac{5 \pi}{6}, \frac{7 \pi}{6}\)