Discovering the Importance and Application of Rolle’s Theorem in Calculus

Rolle’s Theorem

Rolle’s theorem is a fundamental result in calculus that deals with continuous and differentiable functions

Rolle’s theorem is a fundamental result in calculus that deals with continuous and differentiable functions. It states that if a function f(x) is continuous on a 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 (a, b) where the derivative of the function is equal to zero, i.e., f'(c) = 0.

In simpler terms, Rolle’s theorem tells us that if a function is continuous on a closed interval, and its starting and ending points have the same value, then there must be at least one point within the interval where the function has no slope (the derivative is zero).

To understand the significance of this theorem, let’s consider an example. Suppose we have a function f(x) = x^2 – 4x + 3 defined on the interval [1, 4]. We can see that f(1) = f(4) = 0, which satisfies the condition of having equal values at the endpoints. Now we want to find the point(s) where the derivative of the function is zero.

To find the derivative, we differentiate f(x) with respect to x: f'(x) = 2x – 4. Setting f'(x) equal to zero, we have 2x – 4 = 0. Solving this equation for x, we get x = 2. Therefore, according to Rolle’s theorem, there exists at least one point c in the interval (1, 4) where the derivative of the function is zero.

In this example, the point c turns out to be x = 2. We can verify this by calculating f'(2) = 2(2) – 4 = 0. Hence, Rolle’s theorem is satisfied, and it tells us that there must be a stationary point at x = 2 on the graph of the function f(x) = x^2 – 4x + 3.

Rolle’s theorem is primarily used as a stepping stone to prove other important theorems in calculus, such as the mean value theorem. It is a powerful tool that helps us understand the behavior and properties of differentiable functions on closed intervals.

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