Prove Mean Value Theorem: f(x) on I=(a,b)

[mtayab1994]

Homework Statement

1) Let f be a function differentiable two times on the open interval I and a and b two numbers in I

Prove that: $$\exists c\in]a,b[:\frac{f(b)-f(a)}{b-a}=f'(a)+\frac{b+a}{c}f''(c)$$

2) Let f be a function differentiable three times on the open interval I and a and b two numbers in I.

Prove that: $$\exists c\in]a,b[:f(b)=f(a)+(b-a)f'(a)+\frac{(b-a)^{2}}{2}f''(a)+\frac{b-a}{2}f'''(c)$$

The Attempt at a Solution

Any tips on how to start please. Thank you in advance.

 

[micromass]

What did you try?

Are you reminded of some general result or theorem?

 

[mtayab1994]

What did you try?

Are you reminded of some general result or theorem?

Well the theorem states that if a function is continuous on a closed interval [a,b] and is differentiable on the open interval (a,b) then there exists a c in the open interval (a,b) such that.

$$f'(c)=\frac{f(b)-f(a)}{b-a}$$

And then i tried counting the second derivative but i couldn't get anything out of it.

 

[micromass]

Do you know Taylor's theorem?

 

[mtayab1994]

Do you know Taylor's theorem?

Yes i know that it can be solved using taylor's theorem easily, but we need to prove it for the a function differentiable twice and a function differentiable 3 times and then we have to prove taylor's theorem for a function differentiable n times. So I thought that since there exists a C in the open interval (a,b) then f(c) will have to be the mean of the f(b)+f(a) and when i take the derivative of that i get f'(c)=(f'(b)-f'(a))/2, but I don't know what to do from here on.