Skip to main content
Mathovia

Math Practice

Intermediate Value Theorem Practice Problems

For each problem, check that the function is continuous on the closed interval, then compare the endpoint values. A sign change guarantees a root; bisection narrows down where it is.

  1. 1.

    . Find .

    Show full solution

    .

  2. 2.

    For the same , find .

    Show full solution

    .

  3. 3.

    Using and , does the IVT guarantee a root of in ? Type yes or no.

    Show full solution

    is a continuous polynomial and changes sign between 0 and 1, so 0 is hit in between.

  4. 4.

    Can the IVT show that has a root in ? Type yes or no.

    Show full solution

    : no sign change. (It has no real roots at all.)

  5. 5.

    Does the IVT apply to on ? Type yes or no.

    Show full solution

    is not continuous at , which lies inside the interval.

  6. 6.

    on . Which of these values must take: 0, 10 or 20? Type the number.

    Show full solution

    and . Only 10 lies between 1 and 16.

  7. 7.

    Between which two consecutive integers is the root of ? Give the smaller integer.

    Show full solution

    and , so the root is between 1 and 2.

  8. 8.

    Show has a solution by finding for .

    Show full solution

    . Since , there’s a sign change.

  9. 9.

    Apply one bisection step to on . Which half contains the root? Give the left endpoint of that half.

    Show full solution

    and , so the root is in .

  10. 10.

    Apply two bisection steps to starting from . What is the left endpoint of the interval after step 2?

    Show full solution

    Step 1 gives . , so step 2 gives .

  11. 11.

    For , which interval contains a root: [0, 1], [1, 2] or [2, 3]? Give its left endpoint.

    Show full solution

    , : the sign changes on . (, give no change on the others.)

  12. 12.

    has a solution in . Find it to three decimal places.

    Show full solution

    The solution is , which is indeed between 1 and 2.
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Intermediate Value Theorem Practice Problems

These problems follow the intermediate value theorem lesson.

Using the IVT in three checks

  1. Continuity: is continuous on the whole closed interval ?
  2. Endpoints: compute and .
  3. Between: if the target value (often 0) lies between them, some in hits it.

Common mistakes to watch for

  • Skipping the continuity check. The theorem only applies when is continuous on the whole closed interval.
  • Confusing “a root exists” with finding it. The IVT proves existence; bisection or algebra finds the value.
  • Reading no sign change as no root. A function can be positive at both ends and still cross zero twice in between.
  • Forgetting to rearrange first. For , study and look for a sign change.

Frequently Asked Questions

How do I answer the yes or no questions?+

Type yes or no. The question is always whether the intermediate value theorem guarantees the result, not whether a root happens to exist.

Why can the IVT fail even when a root exists?+

The theorem only detects a sign change. A function like x² − 1 on [−2, 2] is positive at both ends but has two roots in between, which the IVT cannot see.

More practice problems

Continuity Practice Problems Types of Discontinuity Practice Problems Polynomial Zeros Practice Problems