(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f : R ---> R be a Continuosly differentiable function such that f(0) = 0 , f(1) = 1 and f'(0) = f'(1) = 0 .

Show that their exists an S in (0, 1) such that |f''(S)| > 4 .

2. Relevant equations

I am unsure if I am even supposed to use the mean value theorm in this problem, but it seems like you have to..

THE MEAN VALUE THEORM:

Let f : [a, b] → R be continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then there exists some R in (a, b) such that

f(b) − f(a) = f'(R)(b − a)

3. The attempt at a solution

Apply derivatives to the mean value theorm above to get:

f'(b) − f'(a) = f''(S)(b − a)

Now, taking the magnitudes, we get

|f'(b) − f'(a)| < |f''(S)||(b − a)|

Now computing, we get

|f'(1) − f'(0)| < |f''(S)||(1 − 0)|

|0 − 0| < |f''(S)|

0< |f''(S)|

Hence

0< |f''(S)|

so indeed, 4< |f''(S)|

How does this sound?

I am very unsure of this...I dont think it is rigourous enough

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Mean Value Theorm type question

**Physics Forums | Science Articles, Homework Help, Discussion**