# Generalized Mean Value Theorem

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• May 28th 2006, 06:58 AM
TexasGirl
Generalized Mean Value Theorem
Let a<b be elements of R, and let f,g:[a,b]-->R be continuous on [a,b] and differentiable on ]a,b[. Assume that f(a) is less than or equal to g(a) and that f'(x)<g'(x) for x element of ]a,b[. Show that f(x)<g(x) for x element of ]a,b].

Can this be done using Generalized Mean Value Theorem? And either way, could somebody give me some pointers?

Thanks a bunch!
• May 28th 2006, 07:19 AM
CaptainBlack
Quote:

Originally Posted by TexasGirl
Let a<b be elements of R, and let f,g:[a,b]-->R be continuous on [a,b] and differentiable on ]a,b[. Assume that f(a) is less than or equal to g(a) and that f'(x)<g'(x) for x element of ]a,b[. Show that f(x)<g(x) for x element of ]a,b].

Can this be done using Generalized Mean Value Theorem? And either way, could somebody give me some pointers?

Thanks a bunch!

Consider $\displaystyle h:[a,b] \rightarrow \manthbb{R}$ such that

$\displaystyle \forall x \in [a,b]\ h(x)=f(x)-g(x)$.

Then $\displaystyle h$ is strictly decreasing in $\displaystyle [a,b]$, and $\displaystyle h(a)\le 0$.

And the rest should be trivial.

RonL
• May 28th 2006, 07:51 AM
TexasGirl
thanks
thanks a bunch =)