Let $f(x)=\sin x-x+\frac {x^3}{6}$ and $g(x)=\cos x-1+\frac {x^2}{2}$ for $x\in \Bbb R$.Then How to prove that $f(x)\ge 0$ for all $x\ge 0$?
From given function it is clear that $f'=g$ and $g'(x)=x-\sin x$ hence $g'(x)\ge0$, hence $g$ is increasing function and hence $f'$. But my question is, if $f' $ is increasing function how to prove that $f(x)\ge 0$ for all $x\ge 0$?
Because now by the definition of the increasing function we obtain: $$f(x)\geq f(0)=0.$$