Twice continuously differentiable function

In summary, the problem is to prove that the absolute value of the derivative of a function f is bounded by a given formula, and the function and its second derivative are also bounded. The conversation discusses possible approaches to solving this problem, including using the Taylor series and redefining the integral limits.
  • #1
Jonas Rist
7
0
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]

and:

[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 don´t know how to go on :yuck:
I´d be glad to get some hints!
Thanks
Jonas

EDIT: Would it make sense to apply the Tayler series here?
 
Last edited:
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  • #2
What is h ?
 
  • #3
Don't include x as the variable in your integral and as a limit, it will only confuse you unnecessarily.
 

What is a twice continuously differentiable function?

A twice continuously differentiable function is a mathematical function that has a continuous first and second derivative. This means that the function is smooth and has no sharp corners or breaks.

What is the difference between a continuous function and a twice continuously differentiable function?

A continuous function is a function that has no sudden jumps or breaks and can be drawn without lifting the pen from the paper. A twice continuously differentiable function is a function that not only has no sudden jumps or breaks, but also has a continuous first and second derivative.

How do you determine if a function is twice continuously differentiable?

To determine if a function is twice continuously differentiable, you need to take the first and second derivative of the function. If both derivatives exist and are continuous, then the function is twice continuously differentiable.

What are some real-life applications of twice continuously differentiable functions?

Twice continuously differentiable functions are commonly used in physics, engineering, and economics to model real-life phenomena such as motion, forces, and economic growth. They are also used in optimization problems to find the maximum or minimum value of a function.

Can a function be twice continuously differentiable but not differentiable?

No, a function cannot be twice continuously differentiable without being differentiable. A function must be differentiable in order to have a first derivative, and it must have a first derivative in order to have a second derivative. Therefore, a function that is twice continuously differentiable must also be differentiable.

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