Question

The length of the longest interval, in which the function $3\,sin\, x - 4\,sin^3\, x$ is increasing, is

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

Answer 1

Answer: (A)

Let $f(x) = 3\, sin \,x - 4sin^3\, x = sin^3\,x$
Since, $sin \,x$ is increasing in the interval $\left[-\frac{\pi}{2}, \frac{\pi }{2}\right]$.
$\therefore -\frac{\pi }{2} \le 3x \le \frac{\pi }{2}$
$\Rightarrow -\frac{\pi }{6} \le x \le \frac{\pi }{6}$
Thus, length of interval $= \left|\frac{\pi }{6} - \left(-\frac{\pi }{6}\right)\right|$
$ = \frac{\pi }{3} $