# What's the name of this theorem ($f>g$ given $f'>g'$ and initial value)?

I need to use this theorem as part of a proof. Basically it goes as follows:

If $f(x_0) > g(x_0)$ and $\frac{d}{dx} f(x) > \frac{d}{dx} g(x)$ for all $x \geq x_0$, then $f(x) > g(x)$ for all $x\geq x_0$.

What's the name of this theorem, if it has one? Or if it doesn't, is there a simple way to prove it that I could include in my proof? Thanks.

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Hint: it follows directly from the mean value theorem. –  mjqxxxx Oct 10 '12 at 2:46

This is commonly called the Racetrack principle.

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Hint

Consider $$h=f-g$$

You have then that $$h'>0$$ and that $$h(x_0)>0$$

Why must then $h$ remain positive for $x\geq x_0$?

Spoiler

If $f'>0$ on some domain $D$; then $f$ is increasing on $D$.

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