Taylor theorem

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Homework Statement


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

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

Then, for and [itex]\delta > 0[/itex], show that holds

[itex]h(L) \leq \frac{2}{\delta}g(L) + \frac{\delta}{2}M[/itex]

Homework Equations





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.....
 

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