Rolles Theorem and Lagranges Theorem

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 \(f\left( x \right)\) be a function defined on [a, b] such that

(i) it is continuous on [a, b]

(ii) it is differentiable on (a, b)

(iii) f (a) = f (b)

Then there exists a real number \(c \in \left( {a,b} \right)\)  such that f '(c) = 0.

The geometrical interpretation of this theorem is quite straightforward. Consider an arbitrary curve y = f (x) and two points x = a and x = b such that f (a) = f (b).

Since A and B are joined by a continuous and differentiable curve, at least one point x = c will always exist in (a, b) where the tangent drawn is horizontal, or equivalently, f '(c) = 0. Convince yourself that no matter what curve joins A and B, as long as it is continuous and differentiable one such c will always exist.

From Rolle’s theorem, it follows that between any two roots of a polynomial f (x) will lie a root of the polynomial f '(x).

The (straightforward) proof of Rolle’s theorem is left as an exercise to the reader.

(B)  LAGRANGE’S MEAN VALUE THEOREM

Let f (x) be a function defined on [a, b] such that

(i)  it is continuous on [a, b]

(ii)  it is differentiable on (a, b).

Then there exists a real number \(c \in \left( {a,b} \right)\) such that

\[f'\left( c \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

To interpret this theorem geometrically, we take an arbitrary function y = f (x) and two arbitrary points x = a and x = b on it

We see that no matter what the curve between R and P is like, as long as it is continuous and differentiable, there will exist a \(c \in \left( {a,\,b} \right)\) such that the tangent drawn at x = c will have a slope equal to tan \(\theta \) i.e, the average slope from x = a to x = b.

For a rigorous proof of LMVT, consider the function

\[g\left( x \right) = f\left( x \right) - \left( {\frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}} \right)x\]

Verify that g(x) satisfies all the three criteria of Rolle’s theorem on [a, b] so that

    \(\qquad g'\left( c \right) = 0\) for at least one \(c \in \left( {a,\,b} \right)\)

or         \(\begin{align}\qquad f'\left( c \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\end{align}\) for at least one \(c \in \left( {a,b} \right)\)

Notice that LMVT is an extension of the Rolle’s theorem. In fact, for f (a) = f (b), LMVT reduces  to the Rolle’s theorem.