Before jumping on to the applications of Rolle’s theorem let us study its definition. Rolle’s theorem simply states that if a function f is differentiable in the open interval (a, b) and continuous in the closed interval [a, b] and if it also attains equal value at two distinct points, i.e., f(a) = f(b), then there is at least one Point c between a and b where the first derivative of the function ( the Slope of the Tangent line to the graph of the function) is zero, i.e., f '(c) = 0. So, it is clear that there exist a point c between a and b, i.e. (a,b) where the tangent line on the function is parallel to the secant drawn between a and b, as shown in figure1.
If we talk about the applications of Rolle’s Theorem, then it plays an important intermediate role or we can say it plays a base in proving many more important theorems like Taylor’s theorem, mean value theorem and extreme value theorem.
Rolle’s theorem is almost interchangeable with Mean value theorem as Mean value theorem states that:
If a function f is differentiable in the open interval (a, b) and continuous in the closed interval [a, b], then there exist a point c at which,
f’(c) = f(b) – f(a) / (b-a).
It is one of the most useful applications of Rolle’s Theorem used in Calculus.
Extreme value theorem which comes under applications of Rolle’s Theorem states that:
If a function f is real valued and is also continuous in the closed interval [a, b], then f(x) must have at least one minimum and one maximum value, that is, there exist points c and d in the closed interval [a, b],
f(c) ≥ f(x) ≥ f(d) for all x in the Domain [a, b].
This theorem plays an important role in determining the maximum and minimum value of the function in certain intervals as earlier we were using critical points for that purpose. So, indirectly we can call it as an application of Rolle’s Theorem.