Question:
Let \( f(x) = (\cos^2 x)(a + \cos x) \). If \( f'\left(\frac{\pi}{3}\right) = 0 \), then the value of \( a \) is equal to
Let \( f(x) = (\cos^2 x)(a + \cos x) \). If \( f'\left(\frac{\pi}{3}\right) = 0 \), then the value of \( a \) is equal to
Show Hint
Plug values early after differentiation to simplify calculations.
Updated On: Apr 21, 2026
- \( \frac{\sqrt{3}}{2} \)
- \( \frac{3}{4} \)
- \( -\frac{3}{4} \)
- \( -\frac{3}{2} \)
- \( -1 \)
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The Correct Option is C
Solution and Explanation
Concept:
Use product rule.
Step 1: Differentiate.
\[ f'(x) = 2\cos x(-\sin x)(a+\cos x) + \cos^2 x(-\sin x) \]
Step 2: Substitute \( x=\frac{\pi}{3} \).
\[ \cos\frac{\pi}{3} = \frac{1}{2}, \quad \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \]
Step 3: Solve equation.
\[ f'(\tfrac{\pi}{3})=0 \Rightarrow a = -\frac{3}{4} \]
Step 1: Differentiate.
\[ f'(x) = 2\cos x(-\sin x)(a+\cos x) + \cos^2 x(-\sin x) \]
Step 2: Substitute \( x=\frac{\pi}{3} \).
\[ \cos\frac{\pi}{3} = \frac{1}{2}, \quad \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \]
Step 3: Solve equation.
\[ f'(\tfrac{\pi}{3})=0 \Rightarrow a = -\frac{3}{4} \]
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