Mean Value Theorem and Rolle's Theorem from "summary" of Differential Calculus by S Balachandra Rao
Mean Value Theorem and Rolle's Theorem are two important theorems in calculus that help us understand the behavior of a function on a given interval. Rolle's Theorem states that if a function f is continuous on a 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 the open interval (a, b) such that f'(c) = 0. In simpler terms, if a function takes the same values at two points on an interval, then the function has a critical point in between where the derivative is zero. The Mean Value Theorem is an extension of Rolle's Theorem and states that if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a). Th...
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