Question:
Show that $f(x)=x^{2}-x \sin x$ is an increasing function on $(0, \pi / 2) ?$
Solution:
We have,
$f(x)=x^{2}-x \sin x$
$f^{\prime}(x)=2 x-\sin x-x \cos x$
Now,
$X \in\left(0, \frac{\pi}{2}\right)$
$\Rightarrow 0 \leq \sin x \leq 1,0 \leq \cos x \leq 1$,
$\Rightarrow 2 x-\sin x-x \cos x>0$
$\Rightarrow f^{\prime}(x) \geq 0$
Hence, $f(x)$ is an increasing function on $\left(0, \frac{\pi}{2}\right)$.