Rolle’s Theorem

Let f be a real valued function that satisfies the following three conditions:

1. f is defined and continuous on the closed interval [a, b]
2. f is differentiable on the open interval (a, b)
3. f(a) = f(b)

Then there exists atleast one point c ∈ (a,b) such that f′(c) = 0

Some Observations

If the object is in the same place at two different instants t = a and t = b then f(a) = f(b) satisfying hypothesis of Rolle’s theorem. Therefore the theorem says that there is some instant of time t = c between a and b where f′(c) = 0 i.e., the velocity is 0 at t = c.

Rolle’s Theorem applied to theory of equations: If a and b are two roots of a polynomial equation f(x) = 0, then Rolle’s Theorem says that there is atleast one root cbetween a and b for f′(x) = 0.

Rolle’s theorem implies that a smooth curve cannot intersect a horizontal line twice without having a horizontal tangent in between.

The converse of Rolle’s Theorem is not true ie., if a function f satisfies f′(c) = 0 for c ∈ (a,b) then the conditions of hypothesis need not hold.