Question

If \(f(\theta) = e^{\sin \theta\), then \(f'(\theta) = 0\) for which value of \(\theta\)}

• A. \(\pi/2\)
• B. \(\pi\)
• C. \(2\pi\)
• D. 0

Hint:

$\frac{d}{d\theta}(e^{\sin\theta}) = e^{\sin\theta}\cos\theta = 0 \implies \cos\theta = 0 \implies \theta = \pi/2$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Finding the stationary points of an exponential-trigonometric composite function by setting its first derivative to zero.
Key Formula or Approach:
\[ \frac{d}{d\theta}[e^{g(\theta)}] = e^{g(\theta)} \cdot g'(\theta) \]

Step 2: Detailed Explanation:
Given the function:
\[ f(\theta) = e^{\sin \theta} \]
Differentiating with respect to \(\theta\) using the chain rule:
\[ f'(\theta) = e^{\sin \theta} \cdot \frac{d}{d\theta}(\sin \theta) = e^{\sin \theta} \cdot \cos \theta \]
Setting \(f'(\theta) = 0\):
\[ e^{\sin \theta} \cdot \cos \theta = 0 \]
Since the exponential function \(e^{\sin \theta} > 0\) for all real \(\theta\):
\[ \cos \theta = 0 \implies \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \dots \]

Step 3: Final Answer:
Therefore, \(f'(\theta) = 0\) for \(\theta = \pi/2\), corresponding to option (A).