So now that we have the mean value theorem what can we do with it? First off, we can tell something that seems intuitively obvious. We know that a constant function has the constant zero function as its derivative. It turns out that these are the only functions with zero derivative.
To see this, let be a differentiable function on so that for all . Let and be any points between and with . Then restricts to a continuous function on the interval which is differentiable on the interior . The differentiable mean value theorem then applies, and it tells us that there is some with . But by assumption this derivative is zero, and so . Since the points were arbitrary, takes the same value at each point in .
What about a function for which on some interval ? Looking at the graph it seems that the slope of all the tangent lines should be positive, and so the function should be increasing. Indeed this is the case.
Specifically we have to show that if for two points in then . Again we look at the restriction of to a continuous function on which is differentiable on . Then the mean value theorem tells us that there is some with . By assumption this quantity is positive, as is , and so . Similarly we can show that if on an interval then the function is decreasing there.