Practice Intermediate Value Theorem in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

If 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 c in (a, b) such that f(c) = N.

A continuous function can't skip values. If you start below a line and end above it, you must cross it somewhere. It's like driving from sea level to a mountaintop—you pass through every elevation in between.

Example 1

easy
Show that f(x) = x^3 - x - 1 has a root in the interval (1, 2).

Example 2

medium
Use the IVT to show that \cos x = x has a solution in (0, 1).

Example 3

easy
Show that h(x) = x^2 - 2 has a root in (1, 2).

Example 4

medium
Can IVT be applied to f(x) = \frac{1}{x} on [-1, 1] to conclude it attains the value 0? Explain.
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