Mathfolis

Intermediate Value Theorem

Unit 1 · Limits & Continuity

The Intermediate Value Theorem

If f is continuous on [a, b] and k is any value strictly between f(a) and f(b), then there exists at least one c in (a, b) with f(c) = k. The most common application is proving a root exists: if f(a) and f(b) have opposite signs, the function must cross zero somewhere in between.

Intermediate Value Theorem

If f continuous on [a,b] and f(a) < k < f(b), then
∃ c∈(a,b) such that f(c)=k\exists\, c \in (a,b) \text{ such that } f(c) = k

Root Existence (Special Case)

If f(a) · f(b) < 0 (opposite signs), then
∃ c∈(a,b) such that f(c)=0\exists\, c \in (a,b) \text{ such that } f(c) = 0
AP Tip: IVT guarantees existence, not uniqueness or location. Always explicitly state: (1) f is continuous on [a, b], (2) the sign change, (3) therefore the root exists by IVT.
Type 1

Proving a Root Exists

Verify continuity, evaluate at both endpoints to find a sign change, then cite the IVT.

Example 1

Use the IVT to prove that f has at least one zero on (1, 2). State all required conditions explicitly.

f(x)=x3+3x−7f(x) = x^3 + 3x - 7

Practice more of this type— AI-generated · infinite problems

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Type 2

IVT with Continuity Analysis

Combine asymptote/continuity analysis with the IVT. First verify continuity on the given interval, then apply IVT.

Example 2

A function f is continuous on [3, 5] with f(3) = −4 and f(5) = 10. Does f have a zero on (3, 5)? Does there exist c in (3, 5) where f(c) = 6?

Practice more of this type— AI-generated · infinite problems

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