1.16 Working with the Intermediate Value Theorem (IVT)

What IVT Says

The Intermediate Value Theorem applies when a function f is continuous on a closed interval [a, b]. It guarantees that for any value k between f(a) and f(b), there exists at least one c in [a, b] such that f(c) = k. It does NOT tell you how many such c's exist or their exact value — only that at least one exists.

Using IVT to Prove Roots Exist

A common use of IVT is proving a function has a zero (root) on an interval. If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then k = 0 is between them, so IVT guarantees some c where f(c) = 0. Always state continuity explicitly — it's the required condition.

Setting Up an IVT Justification

A full justification needs three parts: (1) f is continuous on the given closed interval, (2) evaluate f at both endpoints to show the target value k lies between f(a) and f(b), (3) conclude by IVT that there exists c in (a, b) with f(c) = k. Missing continuity or the endpoint comparison loses credit on the AP exam.

Key terms

Intermediate Value Theorem (IVT)

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c in (a, b) such that f(c) = k.

Required hypothesis for IVT

f must be continuous on the closed interval [a, b] — nothing about differentiability is needed.

Using IVT to show a root exists

Show f is continuous on [a, b] and that f(a) and f(b) have opposite signs; then IVT guarantees some c with f(c) = 0.

What IVT does NOT tell you

It does not say how many values of c satisfy f(c) = k, nor give the exact value of c — only that at least one exists.

Three parts of an IVT justification

1) State f is continuous on [a,b]. 2) Show k is between f(a) and f(b). 3) Conclude by IVT there exists c in (a,b) with f(c) = k.

Try an AP-style question

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

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