PreCalcH Theorems

Immediate Value Theorem IVT: If a function f is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at least once.

Rolle’s Theorem: 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 c in (a, b) such that f'(c) = 0.

Mean Value Theorem MVT: 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 c in (a, b) such that f'(c) = [f(b) - f(a)] / b - a.

Extreme Value Theorem IVT: If a function f is continuous on a closed interval [a, b], then f attains both a maximum and a minimum value, at least once, on that interval.