twice continuously differentiable function

Hello again,

another problem: given: a function

$$f:[0,\infty)\rightarrow\mathbb{R},f\in C^2(\mathbb{R}^+,\mathbb{R})\\$$

The Derivatives

$$f,f''\\$$

are bounded.

It is to proof that

$$\rvert f'(x)\rvert\le\frac{2}{h}\rvert\rvert f\rvert\rvert_{\infty}+\frac{2}{h}\rvert\lvert f''\rvert\rvert_{\infty}\\$$

$$\forall x\ge 0,h>0\\$$

and:

$$\rvert\rvert f'\rvert\rvert_{\infty}\le 2(\rvert\rvert f\rvert\rvert_{\infty})^{\frac{1}{2}}(\rvert\rvert f''\rvert\rvert_{\infty})^{\frac{1}{2}}\\$$

I began like this:

$$f'(x)=\int_{0}^{x}f''(x)dx\Rightarrow$$

$$\rvert f'(x)\rvert\le\rvert\int_{0}^{x}f''(x)dx\rvert\le\int_{0}^{x}\rvert f''(x)\rvert dx$$

But then already I donīt know how to go on
Iīd be glad to get some hints!
Thanks
Jonas

EDIT: Would it make sense to apply the Tayler series here?

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