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

The Correct Option is B

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:

\[ \text{2.} \]