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

Differentiate: $$f'(x) = 12\cos^2 x (-\sin x) + 6\sqrt{3} \cos x (-\sin x) = -6\sin x \cos x (2\cos x + \sqrt{3})$$.

Set $$f'(x) = 0$$:

o $$\sin x = 0 \implies x = \pi$$ (since $$0, 2\pi$$ are excluded).

o $$\cos x = 0 \implies x = \pi/2, 3\pi/2$$.

o $$\cos x = -\sqrt{3}/2 \implies x = 5\pi/6, 7\pi/6$$.

Test for Maxima (Sign change of $$f'(x)$$):

o At $$x = \pi/2$$: $$f'$$ changes from $$-$$ to $$+ \implies$$ Minima.

o At $$x = 5\pi/6$$: $$f'$$ changes from $$+$$ to $$- \implies$$ Maxima.

o At $$x = \pi$$: $$f'$$ changes from $$-$$ to $$+ \implies$$ Minima.

o At $$x = 7\pi/6$$: $$f'$$ changes from $$+$$ to $$- \implies$$ Maxima.

o At $$x = 3\pi/2$$: $$f'$$ changes from $$-$$ to $$+ \implies$$ Minima.

There are 2 points of local maxima ($$5\pi/6$$ and $$7\pi/6$$).

Correct Option: D