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:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

The given function is:

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

Step 1: Taking the derivative:

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

Step 2: Critical points occur when:

$\sin(x) = 0 \quad \text{or} \quad 2\cos(x) + \sqrt{3} = 0.$

Step 3: Solving these equations:

$\sin(x) = 0 \implies x = 0, \pi, 2\pi,$ $\cos(x) = -\frac{\sqrt{3}}{2} \implies x = \frac{5\pi}{6}, \frac{7\pi}{6}.$

Step 4: Checking the interval $(0, 2\pi)$:

The local maxima occur at: $x = \frac{5\pi}{6}, \frac{7\pi}{6}.$

Final Answer:

$\text{2.}$