Question

The length of the longest interval in which the function $3\sin \left(x\right)-4{\sin }^3\left(x\right)$ increases is

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\frac{3\mathrm{\pi }}{2}$
• D. $\mathrm{\pi }$

Answer 1

The correct option is A

$\frac{\mathrm{\pi }}{3}$

Explanation for the correct option

Given function is $3\sin \left(x\right)-4{\sin }^3\left(x\right)$
We know that,

$3\sin \left(x\right)-4{\sin }^3\left(x\right)=\sin \left(3x\right)$

We know that $\sin x$ increases when $x$ goes from $-\frac{\mathrm{\pi }}{2}$ to $\frac{\mathrm{\pi }}{2}$

So, $\sin \left(3x\right)$ increases when $3x$ goes from $-\frac{\mathrm{\pi }}{2}$ to $\frac{\mathrm{\pi }}{2}$ $\Rightarrow$$x$ goes from $-\frac{\mathrm{\pi }}{6}$ to $\frac{\mathrm{\pi }}{6}$

Thus, the length of the longest interval when it increases is $\frac{\mathrm{\pi }}{6}-\left(-\frac{\mathrm{\pi }}{6}\right)=\frac{\mathrm{\pi }}{3}$

Hence, option(A) i.e. $\frac{\mathrm{\pi }}{3}$ is correct.