Question

The value of $a$ for which $f(x) = a\sin x + \dfrac{1}{3}\sin 3x$ has an extremum at $x = \dfrac{\pi}{3}$ is

• A. 1
• B. $-1$
• C. 2
• D. 0

Hint:

For extremum: set $f'(x) = 0$ at the given point and solve for the unknown parameter.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
At an extremum, $f'(x) = 0$.
Step 2: Detailed Explanation:
$f'(x) = a\cos x + \cos 3x$.
At $x = \dfrac{\pi}{3}$: $f'\!\left(\dfrac{\pi}{3}\right) = a\cos\dfrac{\pi}{3} + \cos\pi = \dfrac{a}{2} - 1 = 0 \Rightarrow a = 2$.
Step 3: Final Answer:
$a = 2$.