Practice Intermediate Value Theorem in Math

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

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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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Example 7

medium
Show sin⁡x=x2 has a positive solution.

Example 8

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

Example 9

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

Example 10

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

Example 11

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

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

medium
f is continuous, f(0)=2, f(4)=2. Must f take the value 1 on (0,4)?

Example 20

easy
A continuous function has f(0)=1 and f(3)=7. Is there a c with f(c)=4?
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