The Intermediate Value Theorem

AP Calculus · Axiom Academy

LESSON The Intermediate Value Theorem Prove roots exist without finding them exactly If a continuous function takes on different values at two points, it must take on every value between them. In other words: a continuous function can't "jump over" values—it must hit every value in between. The IVT requires continuity. Discontinuous functions can jump over values. The IVT is perfect for proving that an equation has a solution: If and f(b) > 0, then there's a root between a and b We don't need to find the exact root, just prove it exists Problem: Prove that has a solution between 0 and 1. Since f(0) < 0 < f(1), by IVT, there exists c ∈ (0,1) where f(c) = 0

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