Step 1: Understanding the Concept:
To find the maximum value of a function, we find its critical points by setting the first derivative to zero and analyzing them within the given interval.
Step 2: Detailed Explanation:
Let \( f(x) = \sin x(1 + \cos x) = \sin x + \sin x \cos x = \sin x + \frac{1}{2} \sin 2x \).
Differentiating:
\[ f'(x) = \cos x + \frac{1}{2} \cdot 2 \cos 2x = \cos x + \cos 2x \]
For critical points, set \( f'(x) = 0 \):
\[ \cos x + (2 \cos^{2} x - 1) = 0 \]
\[ 2 \cos^{2} x + \cos x - 1 = 0 \]
Factorizing the quadratic:
\[ (2 \cos x - 1)(\cos x + 1) = 0 \]
This gives \( \cos x = 1/2 \) or \( \cos x = -1 \).
In the given range \( 0 \le x \le \pi/2 \):
\( \cos x = 1/2 \implies x = \pi/3 \).
\( \cos x = -1 \) is not possible in the interval.
Checking the maximum: \( x = \pi/3 \) (which is \( 60^{\circ} \)) is the point of maxima.
Since \( \pi/3 \) is not present in options A, B, or C, the correct choice is "None of these".
Step 3: Final Answer:
The maximum occurs at \( x = \pi/3 \).