- #1

#### playboy

## Homework Statement

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 .

## Homework 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)

## 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 don't think it is rigourous enough