Rolle’s Theorem
Rolle’s Theorem is a fundamental theorem in calculus that is used to find specific points on a curve where the derivative is equal to zero
Rolle’s Theorem is a fundamental theorem in calculus that is used to find specific points on a curve where the derivative is equal to zero. It is named after the French mathematician Michel Rolle.
The statement of Rolle’s Theorem is as follows:
If a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one number c in (a, b) such that f'(c) = 0.
In simpler terms, if a function is continuous on a closed interval and differentiable on the open interval, and it takes the same value at both endpoints, then there must be at least one point within the interval where the derivative is zero.
To apply Rolle’s Theorem, follow these steps:
1. Verify that the function is continuous on the closed interval [a, b]. This means checking that the function does not have any discontinuities, holes, or vertical asymptotes within this interval.
2. Verify that the function is differentiable on the open interval (a, b). This means checking that the derivative exists and is defined for all values of x within this interval.
3. Check if f(a) = f(b), i.e., the function takes the same value at both endpoints. If this condition is not satisfied, then you cannot apply Rolle’s Theorem.
4. If all the above conditions are met, find the derivative f'(x) of the function.
5. Find the values of x where f'(x) = 0. These are the possible candidate points where Rolle’s Theorem guarantees the existence of at least one point c in the interval (a, b) where f'(c) = 0.
6. Finally, you can conclude that there exists at least one point c in the interval (a, b) such that f'(c) = 0, based on Rolle’s Theorem.
Rolle’s Theorem is often used as an intermediate step in proving other theorems, such as the Mean Value Theorem (MVT). It is a powerful tool that helps analyze the behavior of functions and can be used to determine various properties of curves.
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