# Twice continuously differentiable function

by Jonas Rist
Tags: continuously, differentiable, function
 P: 7 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 don´t know how to go on I´d be glad to get some hints! Thanks Jonas EDIT: Would it make sense to apply the Tayler series here?