TITLE: Showing a function is unbounded
QUESTION [7 upvotes]: Let $f(x)=x\cos x+\sin x$. Show that for every $M>0$ and every $k\geq 1$, there is $x_0>M$ such that
$f(x)\geq k$, $\forall x\in [x_0,x_0+\frac{1}{k}]$.
If we replace $x$ by $2\pi n$ then
$|f(2\pi n)=|2\pi n|$.
Can this help?
Can I get a help to start it?

REPLY [2 votes]: Let $x_0=2\pi n$ where $n\in\Bbb Z^+$ is large enough that
$2\pi n(\cos 1)-1\ge k$ and $2\pi n>M.$
If $x\in [x_0,x_0+1/k]$ then $\cos x=\cos (x-2\pi n)\ge \cos 1>0$  so $$f(x)=x\cos x+\sin x\ge 2\pi n\cos x-|\sin x|\ge$$ $$\ge  2\pi n(\cos 1)-|\sin x|\ge$$ $$\ge 2\pi n(\cos 1)-1\ge k.$$