Rolle’s Theorem: Exploring the Fundamental Theorem in Calculus for Differentiable Functions with Zero Derivatives

Rolle’s Theorem

Rolle’s Theorem is a fundamental theorem in calculus named after the French mathematician Michel Rolle

Rolle’s Theorem is a fundamental theorem in calculus named after the French mathematician Michel Rolle. It establishes a condition for a differentiable function to have at least one point where the derivative is equal to zero. The theorem states the following:

If a real-valued function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.

This theorem essentially states that for a continuous and differentiable curve that starts and ends at the same height (i.e., f(a) = f(b)), there must be at least one point where the tangent line is horizontal, or in other words, the derivative f'(x) is equal to zero.

The graphical interpretation of Rolle’s Theorem can be seen as follows: if we imagine a curve that satisfies the conditions of the theorem, the curve must have a flat tangent line at least once between a and b.

The proof of Rolle’s Theorem relies on the Mean Value Theorem (MVT). By applying the MVT to the given function f(x) over the interval [a, b], where f(x) is continuous on [a, b] and differentiable on (a, b), we can conclude that there exists some value c in (a, b) for which:

f'(c) = (f(b) – f(a))/(b – a) = 0.

The equation f'(c) = 0 is derived from the fact that f(b) – f(a) = 0, due to the condition f(a) = f(b).

Rolle’s Theorem is often used as a tool to solve problems in calculus, particularly in proving results related to local extrema (maximum and minimum values) of functions. It can also be used to prove the existence of roots for certain types of equations.

Overall, Rolle’s Theorem serves as an important foundational result in calculus, providing insights into the behavior of differentiable functions and contributing to the development of more advanced concepts in calculus.

More Answers: