Question:
The length of the longest interval, in which $f(x)=3\sin x - 4\sin^3 x$ is increasing, is
The length of the longest interval, in which $f(x)=3\sin x - 4\sin^3 x$ is increasing, is
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Always try to convert expressions using standard identities first.
Updated On: Apr 23, 2026
- $\frac{\pi}{3}$
- $\frac{\pi}{2}$
- $\frac{3\pi}{2}$
- $\pi$
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The Correct Option is A
Solution and Explanation
Concept:
Use identity:
\[
3\sin x - 4\sin^3 x = \sin 3x
\]
Step 1: Rewrite function.
\[ f(x) = \sin 3x \]
Step 2: Differentiate.
\[ f'(x) = 3\cos 3x \]
Step 3: Check increasing condition.
\[ f'(x)>0 \Rightarrow \cos 3x>0 \]
Step 4: Solve inequality.
\[ -\frac{\pi}{2}<3x<\frac{\pi}{2} \] \[ -\frac{\pi}{6}<x<\frac{\pi}{6} \]
Step 5: Find interval length.
\[ = \frac{\pi}{6} - \left(-\frac{\pi}{6}\right) = \frac{\pi}{3} \] Conclusion:
Length = $\frac{\pi}{3}$
Step 1: Rewrite function.
\[ f(x) = \sin 3x \]
Step 2: Differentiate.
\[ f'(x) = 3\cos 3x \]
Step 3: Check increasing condition.
\[ f'(x)>0 \Rightarrow \cos 3x>0 \]
Step 4: Solve inequality.
\[ -\frac{\pi}{2}<3x<\frac{\pi}{2} \] \[ -\frac{\pi}{6}<x<\frac{\pi}{6} \]
Step 5: Find interval length.
\[ = \frac{\pi}{6} - \left(-\frac{\pi}{6}\right) = \frac{\pi}{3} \] Conclusion:
Length = $\frac{\pi}{3}$
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