Find a function $f$

Question

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

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

Please help

For (1) look up Rolle's Theorem. For (2) there's many choices, one of which is a linear y=mx+b.
How can I use Rolle's theorem if we don't have anywhere f(a)=f(b) ?

Ilmari Karonen · Accepted Answer · 2012-10-17 18:55:31Z

up vote 1 down vote accepted

The answer to (1) is probably most easily obtained using the mean value theorem, which is a generalization of Rolle's theorem.

As noted in the comments, there are plenty of suitable examples for (2). In particular, a straight line passing through those two points will do.

answered Oct 17 '12 at 18:55

Ilmari Karonen
8,2461534

Thanks. It's draw. :-) – vesszabo Oct 17 '12 at 19:04

vesszabo · Answer 2 · 2012-10-17 18:54:19Z

up vote 1 down vote

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.
(Hint). Find a linear function.

answered Oct 17 '12 at 18:54

vesszabo
1,493710

+1 for being faster. :) – Ilmari Karonen Oct 17 '12 at 18:56

2 Answers