CHAPTER 17 - Introduction to Mean Value Theorems

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.