Utilizing Rolle’s Theorem to Find Points of Zero Derivative in Calculus

rolle’s theorem

Rolle’s theorem is a fundamental result in calculus that connects the concept of differentiability with the existence of certain points on a function

Rolle’s theorem is a fundamental result in calculus that connects the concept of differentiability with the existence of certain points on a function. It is named after the French mathematician Michel Rolle.

Statement of Rolle’s Theorem:
Let f(x) be a function that satisfies the following conditions:
1. f(x) is continuous on the closed interval [a, b].
2. f(x) is differentiable on the open interval (a, b).
3. f(a) = f(b).

Then, there exists at least one value c in the open interval (a, b) such that f'(c) = 0.

Explanation:
Rolle’s theorem tells us that if a function satisfies the given conditions, and if the function takes the same values at both endpoints (f(a) = f(b)), then there must be at least one point within the interval where the derivative of the function is equal to zero (f'(c) = 0).

Intuitively, if a function starts and ends at the same height within an interval while being continuously and differentiably defined, then at some point between the two endpoints, the function must flatten out and have a horizontal tangent, resulting in a derivative of zero.

Graphical Interpretation:
A graphical representation of Rolle’s theorem consists of a continuous and differentiable function with its graph connecting the points (a, f(a)) and (b, f(b)). If the function satisfies the conditions, there will exist at least one point c within the interval where the tangent line to the graph is horizontal.

Applications:
Rolle’s theorem is often used as a starting point for various mathematical proofs and to prove the existence of solutions to specific problems. It helps establish the existence of critical points or zero gradients within the interval. Additionally, Rolle’s theorem is an essential step in proving other important theorems of calculus, such as the Mean Value Theorem.

Example:
Let’s consider the function f(x) = x^3 – 4x^2 + 3x + 2 on the interval [1, 4].

Conditions satisfied by f(x):
1. The function f(x) is continuous on [1, 4] since it is a polynomial, and polynomials are continuous everywhere.
2. The function f(x) is differentiable on (1, 4) as the derivative of f(x) exists and is defined for all x in (1, 4).
3. f(1) = (1)^3 – 4(1)^2 + 3(1) + 2 = 2 and f(4) = (4)^3 – 4(4)^2 + 3(4) + 2 = 2.

Since the function satisfies all the conditions of Rolle’s theorem, there exists at least one value c in (1, 4) where f'(c) = 0. We can find this point using calculus techniques such as finding derivative f'(x) = 3x^2 – 8x + 3.

Taking the derivative and setting it equal to zero:
3x^2 – 8x + 3 = 0

Factoring or using the quadratic formula, we find:
(3x – 1)(x – 3) = 0

So, x = 1/3 or x = 3. However, the value x = 1/3 lies outside the interval (1, 4).

Hence, using Rolle’s theorem, we conclude that there exists at least one point c in the open interval (1, 4) such that f'(c) = 0. In this case, c = 3 is that point.

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