Bounded derivative

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SemperFi

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How can I show that if a twice continuously differentiable function f:R->R and its second derivative f'' are bounded, the first derivative f' must also be bounded?
It's easy to imagine, but how to prove?

Thank you for your help!
 
SemperFi said:
How can I show that if a twice continuously differentiable function f:R->R and its second derivative f'' are bounded, the first derivative f' must also be bounded?
It's easy to imagine, but how to prove?

Thank you for your help!
Click to expand...

What is the definition of a bounded function?
 
SemperFi said:
How can I show that if a twice continuously differentiable function f:R->R and its second derivative f'' are bounded, the first derivative f' must also be bounded?
It's easy to imagine, but how to prove?
Click to expand...

What about \(\displaystyle f(x)=x^2+x+1~?\)
 
SemperFi said:
...if a twice continuously differentiable function f:R->R and its second derivative f'' are bounded...
Click to expand...
pka said:
What about \(\displaystyle f(x)=x^2+x+1~?\)
Click to expand...
I could be misreading (or misunderstanding) things, but I think both the original function f(x) and also the second derivative are to be bounded. I don't think a quadratic would qualify. ;)
 
Subhotosh Khan said:
What is the definition of a bounded function?
Click to expand...

A function is bounded if there is a real K, such that |f(x)|<K for all real x.


stapel said:
I could be misreading (or misunderstanding) things, but I think both the original function f(x) and also the second derivative are to be bounded. I don't think a quadratic would qualify. ;)
Click to expand...

You are right. Both, the original function f and the second derivative are bounded.
 
Can you use something similar to #19 here?
 
Last edited:
SemperFi said:
Thank you, this is a great approach!
Click to expand...
The above comment is in reference to work assigned to a fourth year/first year graduate course at Georgia Tech.
But from my point of view, it was posted in a forum dedicated to undergraduate calculus. I know an advanced underrate/graduate question when I see one. I knew that this was a standard measure theory question. But as a calculus question about a function with an unbounded domain, it made no sense. Given its placement, I read it in the only way possible to make sense of it.

I ask all of you, why do you expect us to do otherwise ? Why not require posting in a correct forum?
 
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