Understanding Rolle’s Theorem: Exploring a Fundamental Result in Calculus

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 provides conditions under which a differentiable function will have at least one point where its derivative is zero.

The theorem states that if a function f(x) is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in the open interval (a, b) where f'(c) = 0. In other words, if a function is continuous and takes on the same value at its endpoints, then it has a critical point (where the derivative is zero) somewhere in between.

To better understand the theorem, let’s look at an example:

Suppose we have the function f(x) = x^2 – 4x + 3 defined on the interval [0, 3].

Step 1: Checking the conditions of Rolle’s theorem
– Continuity: f(x) is a polynomial, so it is continuous on the interval [0, 3].
– Differentiability: f(x) is also differentiable as it is a polynomial.

Step 2: Checking if f(a) = f(b)
– f(0) = (0)^2 – 4(0) + 3 = 3
– f(3) = (3)^2 – 4(3) + 3 = 0

Since f(0) is not equal to f(3), these values are different, and we cannot apply Rolle’s theorem to this example.

If the condition f(a) = f(b) was satisfied, we could conclude that somewhere in the interval (0, 3) there exists at least one point where the derivative of the function f(x) equals zero.

Rolle’s theorem is a special case of the more general Mean Value Theorem (MVT). The difference is that the MVT requires the function to be not only continuous and differentiable throughout the interval but also strictly increasing or decreasing. Rolle’s theorem relaxes this condition by only requiring the function to take on the same value at the endpoints.

Please note that Rolle’s theorem is a theoretical result that provides existence guarantees but does not provide information about the location of the critical point(s). It’s essential to use other methods, such as the first or second derivative test or graphing techniques, to determine the exact location of critical points and analyze the behavior of the function.

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