Question

\(\max_{0 \leq x \leq \pi} \left( 16 \sin\left(\frac{x}{2}\right) \cos^3\left(\frac{x}{2}\right) \right)\) is equal to:

• A. 2
• B. \(3\sqrt{3}\)
• C. \(4\sqrt{3}\)
• D. \(6\sqrt{3}\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The function can be simplified using double angle trigonometric identities. Then, use the derivative test to find the maximum value on the given interval.

Step 2: Key Formula or Approach:
1. \(2 \sin \theta \cos \theta = \sin 2\theta\).
2. Express as \(y = 4 \sin x (1 + \cos x)\) or use substitution \(u = \cos(x/2)\).

Step 3: Detailed Explanation:
\(y = 16 \sin(x/2) \cos^3(x/2) = 8 \sin(x) \cos^2(x/2) = 4 \sin(x) (1 + \cos x) = 4\sin x + 2\sin 2x\).
\(y' = 4\cos x + 4\cos 2x = 4(\cos x + 2\cos^2 x - 1) = 4(2\cos x - 1)(\cos x + 1)\).
For maxima, \(y' = 0 \implies \cos x = 1/2\) or \(\cos x = -1\).
Since \(0 \leq x \leq \pi\), \(x = \pi/3\).
Max value \(= 4\sin(\pi/3) + 2\sin(2\pi/3) = 4(\sqrt{3}/2) + 2(\sqrt{3}/2) = 2\sqrt{3} + \sqrt{3} = 3\sqrt{3}\).

Step 4: Final Answer:
The maximum value is \(3\sqrt{3}\).