In this section we will deal with some straight forward but quite useful applications of derivatives. We start with the Rolle’s theorem, a simple but powerful theorem having a lot of practical importance.
(A) ROLLE’S THEOREM
Let be a function defined on such that
(i) it is continuous
(ii) it is differentiable on
(iii)
Then there exists a real number such that .
The geometrical interpretation of this theorem is quite straightforward. Consider an arbitrary curve and two points and such that .
Since and are joined by a continuous and differentiable curve, at least one point will always exist in where the tangent drawn is horizontal, or equivalently, . Convince yourself that no matter what curve joins and , as long as it is continuous and differentiable one such will always exist.
From Rolle’s theorem, it follows that between any two roots of a polynomial will lie a root of the polynomial .
The (straightforward) proof of Rolle’s theorem is left as an exercise to the reader.
(B) LAGRANGE’S MEAN VALUE THEOREM
Let be a function defined on such that
(i) it is continuous on
(ii) it is differentiable on .
Then there exists a real number such that
To interpret this theorem geometrically, we take an arbitrary function and two arbitrary points and on it
We see that no matter what the curve between and is like, as long as it is continuous and differentiable, there will exist a such that the tangent drawn at will have a slope equal to i.e, the average slope from to .
For a rigorous proof of , consider the function
Verify that satisfies all the three criteria of Rolle’s theorem on so that
for at least one
 
or
 
for at least one

Notice that is an extension of the Rolle’s theorem. In fact, for , reduces to the Rolle’s theorem.