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Math Practice

Rolle's Theorem Practice Problems

Check that f is continuous on [a, b], differentiable on (a, b) and that f(a) = f(b). Then solve f′(c) = 0 and keep only the values strictly between a and b.

  1. 1.

    Find the value of guaranteed by Rolle's theorem for on .

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    . gives .

  2. 2.

    Find for on .

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    . gives .

  3. 3.

    Find for on .

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    and . gives .

  4. 4.

    Find for on . You can type pi.

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    . gives .

  5. 5.

    Find every for on . Separate answers with commas; you can type sqrt( ).

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    . gives , both inside .

  6. 6.

    Find for on .

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    , and is differentiable on . gives ; only is in .

  7. 7.

    Find for on .

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    . gives , so . Inside only .

  8. 8.

    Find every for on .

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    . gives , all inside .

  9. 9.

    Find for on .

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    and . gives or . Since 0 is an endpoint, .

  10. 10.

    Find every for on .

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    . Expanding, , so . The quadratic formula gives and , both inside .

  11. 11.

    Find for on .

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    . gives , so .

  12. 12.

    How many real solutions does have?

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    Let . and , so there is at least one root. , so by Rolle's theorem there can't be two roots (between them would have to be 0). Exactly one.
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MPhil in Mathematics, Founder of Mathovia
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Rolle’s Theorem Practice Problems

These problems follow the Rolle’s theorem lesson. Most ask for the value of ; the last one uses the theorem to count the roots of an equation.

The method

  1. Check that is continuous on and differentiable on .
  2. Check that .
  3. Solve .
  4. Keep only the solutions strictly between and .

Common mistakes to watch for

  • Keeping an endpoint as a value of .
  • Missing a value when has several solutions inside the interval.
  • Using the theorem without checking the conditions, such as on a function with a corner.

Frequently Asked Questions

How should I type my answer?+

List every value of c separated by commas, in any order. You can type pi and sqrt( ), for example pi/2 or 2/sqrt(3).

Can c be one of the endpoints?+

No. Rolle's theorem places c strictly between a and b, so discard any solution of f′(c) = 0 that equals a or b.

More practice problems

Extreme Value Theorem Practice Problems Critical Points Practice Problems Intermediate Value Theorem Practice Problems