# Find a function $f$

I am having a problem with this exercise. Please help.

Is it possible to find a function $f$ with a continuous derivative $f'$, such that $f'(x)>0$ and

1. $f(0)=1,\;f(1)=0$,
2. $f(0)=-1,\;f(1)=0$?

If yes give an example, and if not, show why

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For (1) look up Rolle's Theorem. For (2) there's many choices, one of which is a linear y=mx+b. –  coffeemath Oct 17 '12 at 18:33
How can I use Rolle's theorem if we don't have anywhere f(a)=f(b) ? –  Carpediem Oct 17 '12 at 18:40
How do I use it then here ? –  Carpediem Oct 17 '12 at 18:49

1. By Lagrange mean -value theorem there exists $0<\tau<1$ such that $$f'(\tau)=\frac{f(1)-f(0)}{1-0}=-1,$$ which is a contradiction. So there is no such function.