calculus - Use the Mean Value Theorem to prove inequality

f is a continuous function defined on [a,b] and differentiable on ]a,b[ with f'(x)>0 on ]a,b[.

Use the mean value theorem to prove that for any x, y, all real [a,b], if y > x then f(y)>f(x)

I understand what the MVT is and what the collalories are, but I just can't figure out how to do these types of questions!

Thanks!

Answer

From the MVT, if $y \gt x$, there exists some $c$ in interval $(x,y)$ such that slope of the chord joining $x$ and $y$ equals $f'(c)$.

So, $\frac{f(y)-f(x)}{y-x}$ should equal $f'(c)$. Since it is given that the derivative is positive throughout the interval, $f'(c) \gt 0$.
It should follow that $f(y)-f(x)$ is positive since $y-x$ is negative.
Thus, $f(y) \gt f(x)$ if $y \gt x$.