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Extreme Value Theorem Practice Problems

Find the critical points inside the interval, evaluate f there and at both endpoints, then compare. Each answer is the maximum or minimum value, not the x-value, unless the question asks where.

  1. 1.

    Find the absolute maximum value of on .

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    at . Candidates: , , . The absolute maximum is 8.

  2. 2.

    Find the absolute minimum value of on .

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    at . Candidates: , , . The absolute minimum is .

  3. 3.

    Find the absolute maximum value of on .

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    is never 0, so only the endpoints count: and . The absolute maximum is 8.

  4. 4.

    Find the absolute minimum value of on .

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    at ; only is in . Candidates: , , . The minimum is .

  5. 5.

    Find the absolute maximum value of on .

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    at . Candidates: , , , . The maximum value is 0, reached at and .

  6. 6.

    Find the absolute maximum value of on .

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    at . Candidates: , , . The maximum is 8.

  7. 7.

    Find the absolute maximum value of on . A fraction or a decimal to two places is fine.

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    at in the interval. Candidates: , , . The maximum is .

  8. 8.

    Find the absolute maximum value of on . You can type sqrt( ).

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    at . Candidates: , , . The maximum is .

  9. 9.

    Find the absolute maximum value of on . You can type e.

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    at . Candidates: , , . The maximum is .

  10. 10.

    Find the absolute maximum value of on .

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    is never 0 and undefined at . Candidates: , , . The maximum is 4, at the left endpoint.

  11. 11.

    Find the absolute minimum value of on . Give a decimal rounded to 3 places.

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    at . Candidates: , , . The minimum is .

  12. 12.

    At which x-values does reach its absolute maximum on ? List every one, separated by commas.

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    , so the critical points are 1 and 3. Candidates: , , , . The maximum value 6 is reached at and .
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Extreme Value Theorem Practice Problems

These problems follow the extreme value theorem lesson. Each one asks for an absolute maximum or minimum on a closed interval, using the closed interval method.

The method

  1. Find the critical points of that lie inside .
  2. Evaluate at those critical points.
  3. Evaluate at the endpoints and .
  4. The largest value is the absolute maximum; the smallest is the absolute minimum.

Common mistakes to watch for

  • Skipping the endpoints, where the absolute extremes often are.
  • Including critical points outside the interval.
  • Giving the x-value when the question asks for the maximum or minimum value.

Frequently Asked Questions

Should I give the x-value or the function value?+

Give the value of the function, the y-value, unless the question asks where the maximum or minimum happens. For example, the absolute maximum of x² − 4x + 3 on [0, 5] is 8, reached at x = 5.

How should I type exact answers?+

Fractions like 10/3, sqrt(2) and 1/e are accepted, and so are decimals rounded as the question asks.

More practice problems

Critical Points Practice Problems Continuity Practice Problems Power Rule Practice Problems