# Prove that f(100) < 100

1. Aug 26, 2011

### flyingpig

1. The problem statement, all variables and given/known data

For f(0) = 0, and that f'(x) = $$\frac{1}{1 + e^{-f(x)}}$$, prove that f(100) < 100

3. The attempt at a solution

I did

$$\f(100) = \int_{0}^{100} \frac{dx}{1 + e^{-f(x)}}$$

Unfortunately, I got f(100) back...

2. Aug 26, 2011

### fzero

3. Aug 26, 2011

### micromass

Staff Emeritus
That's already ok. Can you prove now that the integral must be <100??

First, can you prove that

$$\frac{1}{1+e^{-f(x)}}\leq 1$$

4. Aug 26, 2011

### Ray Vickson

For y > = 0 we have 1/(1+exp(-y)) = exp(y)/[1+exp(y)] <= 1.

RGV