Jonas Rist
May24-04, 04:25 PM
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 :yuck:
Iīd be glad to get some hints!
Thanks
Jonas
EDIT: Would it make sense to apply the Tayler series here?
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 :yuck:
Iīd be glad to get some hints!
Thanks
Jonas
EDIT: Would it make sense to apply the Tayler series here?