1.16 The Intermediate Value Theorem (IVT)

The theorem

IVT: if f is continuous on [a, b] and N is any value between f(a) and f(b), then there is at least one c in (a, b) with f(c) = N. Continuity is required.

Guaranteeing a root

The most common use: if f(a) and f(b) have opposite signs, then f has a zero somewhere between them — a value where f(c) = 0.

Key terms

Intermediate Value Theorem

Continuous f on [a,b] attains every value between f(a) and f(b).

Continuity requirement

The IVT needs f continuous on the closed interval.

Guarantees a root

Opposite signs at the endpoints force a zero.

N between endpoints

Some c gives f(c) = N for any N between f(a), f(b).

Common use

Showing an equation has a solution on an interval.

Try an AP-style question

f is continuous on [1, 4], f(1) = −3, and f(4) = 5. Which must be true?

1. f is increasing on the interval [1, 4]
2. f(c) = 0 for exactly one c in (1, 4)
3. f(c) = 0 for at least one c in (1, 4)
4. f(c) = 6 for some c in (1, 4)