differentiability and continuity problem

#1 Suppose that the differentiable function satisfies and . How large can possibly be?(Done)

If , then show that for .

How to approach this?

#2 Assume is continuous on , and for any , there exists y\in[a,b] such that . Prove that there exists a point such that

Re: differentiability and continuity problem

Re: differentiability and continuity problem

I have just learned about Mean Value Theorem, is it similar to Lagrange Theorem?

btw, if , we can just show that , but not

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** #2 Assume

is continuous on

, and for any

, there exists y\in[a,b] such that

. Prove that there exists a point

such that

__Hint__: is continuous on .

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** I have just learned about Mean Value Theorem, is it similar to Lagrange Theorem?

Yes, it is another name.

Quote:

btw, if

, we can just show that

, but not

You can prove that for all so in **etc** ...

Re: differentiability and continuity problem

Quote:

Originally Posted by

**FernandoRevilla** Yes, it is another name.

You can prove that

for all

so

in

**etc** ...

Is it because is differentiable, then if , we get ?

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** Is it because

is differentiable, then if

, we get

?

Right.

Re: differentiability and continuity problem

Quote:

Originally Posted by

**FernandoRevilla** __Hint__:

is continuous on

.

What's the next step when is continuous on ?

Can i simply state that there exists such that

Re: differentiability and continuity problem

Re: differentiability and continuity problem

I saw another problem which is similar to #1

Suppose that and . How large can possibly be? (Done)

If , then show that for

The question i want to ask is how can i prove if i don't know is differentiable ?

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** The question i want to ask is how can i prove

if i don't know

is differentiable ?

You can't, there are infinite functions satisfying and .