## Rolle’s Theorem

### Rolle’s Theorem is a fundamental result in calculus that is used to find a specific value of a function where its derivative is zero in a specified interval

Rolle’s Theorem is a fundamental result in calculus that is used to find a specific value of a function where its derivative is zero in a specified interval. It establishes a condition for the existence of at least one point on a function’s graph where the tangent line is horizontal.

The formal statement of Rolle’s Theorem is as follows:

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 value c in the open interval (a, b) such that f'(c) = 0.

In simpler terms, if a function is continuous and has equal values at its endpoints, then there must exist a point within the interval where its derivative is zero.

Here’s an example to illustrate the concept of Rolle’s Theorem:

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

First, we check if the given function satisfies the conditions of Rolle’s Theorem. We need to ensure that the function is continuous on the closed interval [0, 3] and differentiable on the open interval (0, 3).

Since f(x) is a polynomial function, it is continuous and differentiable on its entire domain, which includes the interval [0, 3].

Next, we need to verify if f(0) = f(3). Evaluating the function at the endpoints: f(0) = 3 and f(3) = 3. Since f(0) = f(3), the condition for Rolle’s Theorem is met.

Now, we can apply Rolle’s Theorem to find a value c such that f'(c) = 0. We know that f(x) = x^2 – 4x + 3. Taking the derivative, f'(x) = 2x – 4.

To find the value of c, we set f'(x) = 0 and solve for x:

2x – 4 = 0

2x = 4

x = 2

Therefore, according to Rolle’s Theorem, there exists at least one value c in the open interval (0, 3) such that f'(c) = 0. In this case, c = 2.

It is important to note that while Rolle’s Theorem guarantees the existence of a point where the derivative is zero, it does not provide information about other possible roots of the derivative or the behavior of the function outside the given interval.

Rolle’s Theorem is a useful tool in calculus and serves as the basis for several other important theorems, such as the Mean Value Theorem.

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