prove $f$ is a constant Lets's say we have a differentiable function
$f:[a,b]\to \mathbb{R}$
with
$f^\prime\equiv0$
How do I show that 
$f\equiv C$
by using the mean value theorem?

One approach is to prove by contradiction (or contrapositive).
Suppose that $f$ is not constant. Then, there exists a $b' \in [a,b]$ such that $f(b') \neq f(a)$.  Apply the mean value theorem on the interval $[a,b']$.  What do you conclude?
