Rolle's Theorem is a mathematical theorem that states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if the function's value at the endpoints of the interval is equal (f(a) = f(b)), then there exists at least one point c in the open interval (a, b) where the derivative of the function is equal to 0 (f'(c) = 0).
Rolle's Theorem is used to prove exactly one real root by showing that there is only one point in the interval (a, b) where the derivative of the function is equal to 0. This means that the function has a critical point at that point, and since the function is continuous, it must either have a local maximum or a local minimum at that point. This local extremum can only occur at one point, thus proving that there is only one real root.
The conditions for using Rolle's Theorem are that the function must be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). Additionally, the function's value at the endpoints of the interval must be equal (f(a) = f(b)).
No, Rolle's Theorem can only be used to prove exactly one real root. This is because the theorem states that there is at least one point in the interval (a, b) where the derivative of the function is equal to 0. This means that there can be more than one point where the derivative is equal to 0, but there can only be one local extremum for the function, and therefore only one real root.
Yes, there are limitations to using Rolle's Theorem to prove exactly one real root. The function must satisfy the conditions of the theorem, which means that it must be continuous and differentiable on the specified interval. If the function does not meet these conditions, then Rolle's Theorem cannot be applied. Additionally, the theorem can only be used for univariate functions (functions with one independent variable). It cannot be applied to multivariate functions.