Let f ( x ) = 4 cos 3 x + 3 3 cos 2 x - 10 . The number of points of local maxima of f in the interval ( 0 , 2) is: [2024]

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Solution

Q.
Let $f (x) = 4 \cos^{3} x + 3 \sqrt{3} \cos^{2} x - 10$ . The number of points of local maxima of $f$ in the interval $(0, 2 π)$ is: [2024]

1 2

2 3

3 4

4 1

Ans.
(1)

$f (x) = 4 \cos^{3} x + 3 \sqrt{3} \cos^{2} x - 10$

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

$= - 6 \sin x \cos x [2 \cos x + \sqrt{3}]$

Critical points are $x = 0, \frac{π}{2}, π - \frac{π}{6}, π + \frac{π}{6}, π + \frac{π}{2}, π, 2 π$

Sign of $f (x)$

Points of local maxima $= \frac{5 π}{6}, \frac{7 π}{6}$

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