Ivt

Math

Definition

The Intermediate Value Theorem states that if a function f is continuous on the closed interval [a, b] and N is any value between f(a) and f(b), then there exists at least one value c in (a, b) such that f(c) = N. It is commonly used to prove a root exists.

Examples

  • If f(1) = −2 and f(3) = 5 and f is continuous, then by IVT there must be some c between 1 and 3 where f(c) = 0
  • Proving that x³ − x − 1 = 0 has a solution between x = 1 and x = 2 since f(1) = −1 and f(2) = 5
Key Fact

If f is continuous on [a, b] and N is between f(a) and f(b), then f(c) = N for some c in (a, b)

Related Terms

Continuity Mean value theorem Squeeze theorem Zeros and end behavior

Study This Concept

Practice IVT with free review games in these units:

AP Calculus AB Unit 1: Limits and Continuity