Practice Intermediate Value Theorem in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

If ff is continuous on the closed interval [a,b][a, b] and NN is any value between f(a)f(a) and f(b)f(b), then there exists at least one cc in (a,b)(a, b) such that f(c)=Nf(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.

Showing a random 20 of 50 problems.

Example 1

easy
State the two hypotheses required to apply the IVT on [a,b][a,b].

Example 2

easy
If ff is continuous and f(2)=f(5)=3f(2) = f(5) = 3, does IVT guarantee ff takes the value 10 on [2,5][2,5]?

Example 3

medium
Show the equation x=cos⁡(x)x = \cos(x) has a solution in [0,1][0,1].

Example 4

easy
Let f(x)=x2−5f(x)=x^2-5. Use IVT to show ff has a root in (2,3)(2,3).

Example 5

medium
Show f(x)=ex+x=2f(x) = e^x + x = 2 has exactly one solution, locating an interval.

Example 6

medium
Show f(x)=x3+x−1f(x) = x^3 + x - 1 has a root in (0,1)(0, 1).

Example 7

medium
Show sin⁡x=x2\sin x = \frac{x}{2} has a positive solution.

Example 8

hard
Show that the polynomial p(x)=x5−4x3+x−1p(x)=x^5-4x^3+x-1 has at least three real roots.

Example 9

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

Example 10

hard
Show that every continuous function f:[0,1]→[0,1]f:[0,1] \to [0,1] has a fixed point.

Example 11

medium
Show that f(x)=x4−3x−1f(x)=x^4-3x-1 has a root in (1,2)(1,2).

Example 12

challenge
Show f(x)=x3−3x+1f(x) = x^3 - 3x + 1 has three real roots by locating sign changes.

Example 13

medium
Given continuous ff with f(1)=2,f(2)=−1,f(3)=4f(1)=2, f(2)=-1, f(3)=4, what is the minimum number of roots IVT guarantees on [1,3][1,3]?

Example 14

easy
True or false: IVT guarantees exactly one cc with f(c)=Nf(c)=N.

Example 15

medium
Prove every odd-degree polynomial p(x)p(x) has at least one real root.

Example 16

easy
Why must ff be continuous on the CLOSED interval [a,b][a,b] for IVT?

Example 17

medium
A continuous ff maps [0,1][0,1] into [0,1][0,1]. Show ff has a fixed point (f(c)=cf(c)=c).

Example 18

easy
A continuous function has f(−1)=2f(-1)=2 and f(2)=2f(2)=2. Does IVT guarantee f(c)=0f(c)=0 for some c∈(−1,2)c \in (-1,2)?

Example 19

medium
ff is continuous, f(0)=2f(0)=2, f(4)=2f(4)=2. Must ff take the value 11 on (0,4)(0,4)?

Example 20

easy
A continuous function has f(0)=1f(0) = 1 and f(3)=7f(3) = 7. Is there a cc with f(c)=4f(c) = 4?
← Back to Intermediate Value Theorem Worked Examples Formula Explained