Question

Show that $f\left(x\right)=x^2-x\sin$ ,x is an increasing function on $(0,\pi /2)$.

Answer 1

If $f\left(x\right)=x^2-x\sin x$, then

$f^{\prime }\left(x\right)=2x-\sin x-x\cos x=x(2-cosx)-\sin x$

But for all x in $(0,\frac{\pi }{2}),$ $0<\cos x<1$ and $0<x<\sin x$, so

$f^{\prime }\left(x\right)>x-\sin x>0$

which implies that f is strictly increasing on $(0,\frac{\pi }{2})$