Question

If kπ is the length of the largest interval in which the function f(x) = 3sin x - 4sin³x is increasing, then k = _________________.

Answer 1

The given function is f(x) = 3sin x − 4sin³x.

f(x) = 3sin x − 4sin³x = sin3x

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=3\cos 3x$

For f(x) to be increasing,

$f\text{'}\left(x\right)\ge 0$

$\Rightarrow 3\cos 3x\ge 0$

$\Rightarrow \cos 3x\ge 0$

$\Rightarrow -\frac{\mathrm{\pi }}{2}\le 3x\le \frac{\mathrm{\pi }}{2}$

$\Rightarrow -\frac{\mathrm{\pi }}{6}\le x\le \frac{\mathrm{\pi }}{6}$

∴ Length of the largest interval in which the given function f(x) is increasing = $\frac{\mathrm{\pi }}{6}-\left(-\frac{\mathrm{\pi }}{6}\right)=\frac{2\mathrm{\pi }}{6}=\frac{\mathrm{\pi }}{3}$

It is given that, the length of the largest interval in which the function f(x) = 3sin x − 4sin³x is increasing is k$\mathrm{\pi }$.

$\therefore k\mathrm{\pi }=\frac{\mathrm{\pi }}{3}$

$\Rightarrow k=\frac{1}{3}$

Thus, the value of k is $\frac{1}{3}$.


If kπ is the length of the largest interval in which the function f(x) = 3sin x − 4sin³x is increasing, then k = $\underline{\frac{1}{3}}$.