Register to reply

Twice continuously differentiable function

by Jonas Rist
Tags: continuously, differentiable, function
Share this thread:
Jonas Rist
May24-04, 04:25 PM
P: 7
Hello again,

another problem: given: a function

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

The Derivatives

[tex] f,f''\\ [/tex]

are bounded.

It is to proof that

[tex] \rvert f'(x)\rvert\le\frac{2}{h}\rvert\rvert f\rvert\rvert_{\infty}+\frac{2}{h}\rvert\lvert f''\rvert\rvert_{\infty}\\ [/tex]

[tex]\forall x\ge 0,h>0\\ [/tex]


[tex] \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}}\\ [/tex]

I began like this:

[tex] f'(x)=\int_{0}^{x}f''(x)dx\Rightarrow [/tex]

[tex] \rvert f'(x)\rvert\le\rvert\int_{0}^{x}f''(x)dx\rvert\le\int_{0}^{x}\rvert f''(x)\rvert dx [/tex]

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

EDIT: Would it make sense to apply the Tayler series here?
Phys.Org News Partner Science news on
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
May26-04, 11:04 AM
P: 117
What is h ?
matt grime
May26-04, 11:25 AM
Sci Advisor
HW Helper
P: 9,396
Don't include x as the variable in your integral and as a limit, it will only confuse you unnecessarily.

Register to reply

Related Discussions
Continuously Differentiable Piecewise Function? Calculus & Beyond Homework 5
Non continuously differentiable but inner product finite Set Theory, Logic, Probability, Statistics 2
Differentiable function Calculus & Beyond Homework 2
Why is a function not differentiable at a point? Calculus 29
How to tell if a function is differentiable or not Calculus 26