# Taylor theorem

## Homework Statement

$f(x)$ 2 times differentiable function on $(0, \infty)$, and $\lim\limits_{x \rightarrow \infty} f(x)=0$. there is a $M$ such that $M=\sup\limits_{x>0}\vert f^{\prime\prime}(x) \vert$. And also for $L>0$

$g(L)=\sup\limits_{x>L}\vert f(x) \vert$, and $h(L)=\sup\limits_{x>L}\vert f^{\prime}(x) \vert$.

Then, for and $\delta > 0$, show that holds

$h(L) \leq \frac{2}{\delta}g(L) + \frac{\delta}{2}M$

## The Attempt at a Solution

I don't know how should I start,
but I think we need to use Taylors theorem,
can you help.....