# Thread: Mean value theorem ?

1. ## Mean value theorem ?

Help me please with this one, I think it's refers to Mean value theorem, but not sure...

Let us, derivative of a function$\displaystyle f(x)$ - is continuous and $\displaystyle f(0)=1$. Also $\displaystyle \forall x>0: f'(x)>f(x)$.

Prove that $\displaystyle \forall x>0: f(x)>e^{x}$

Thanks!

2. Here goes some hints. Before I solve them for you, try yourself to do it first:-

(i) Define a function $\displaystyle g(x) = e^{-x}f(x)$

(ii) Find $\displaystyle g'(x)$ and show that $\displaystyle g(x)$ is an increasing function.

(iii) From the criterion $\displaystyle g(x)>g(0)$ for all $\displaystyle x>0$, get the desired result.

Note: No use of Mean Value Theorem whatsoever.

3. Great!
Thank you, but the lecturer wanted a solution using Mean Value Theorem...
Is there a solution?

4. ## Thanks

Great!
Thank you, but the lecturer wanted a solution using Mean Value Theorem...
Is there a solution?

5. Originally Posted by sinichko
Great!
Thank you, but the lecturer wanted a solution using Mean Value Theorem...
Is there a solution?
I don't think so.