Can You Prove f(100) Is Less Than 100 Given f(0) = 0 and a Specific Derivative?

[flyingpig]

Homework Statement

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

The Attempt at a Solution

I did

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

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

 

[fzero]

You should try to apply the Mean Value Theorem.

 

[micromass]

I did

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

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

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

 

[Ray Vickson]

Homework Statement

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

The Attempt at a Solution

I did

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

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

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

RGV