Skip to main content
Mathovia

Math Practice

First Derivative Test Practice Problems

Find the critical points, test the sign of f′ in each interval, and read the chart: + to − is a local max, − to + is a local min, and no change is neither. Read each question to see whether it wants an x-value or a function value.

  1. 1.

    At what x-value does have a local minimum?

    Show full solution

    is negative for and positive for . That's − to +, so the local minimum is at .

  2. 2.

    Find the local maximum value of .

    Show full solution

    changes from + to − at . The local maximum value is .

  3. 3.

    At what x-value does have a local maximum?

    Show full solution

    . The sign is + on , − on , + on . The + to − change is at .

  4. 4.

    Find the local minimum value of .

    Show full solution

    changes from − to + at . The local minimum value is .

  5. 5.

    Find the local maximum value of .

    Show full solution

    . The sign changes from + to − at , so the local maximum is .

  6. 6.

    Find every x-value where has a local minimum. Separate them with commas.

    Show full solution

    . The signs are − on , + on , − on , + on . The − to + changes are at and ; is a local maximum.

  7. 7.

    Find every x-value where has a local extremum. Separate them with commas.

    Show full solution

    . Since , the sign follows : − on both sides of 0, + after 1. So is neither, and the only local extremum is a minimum at .

  8. 8.

    Find the local minimum value of .

    Show full solution

    . The signs are + on , − on and , + on . The local minimum is . (The local maximum is .)

  9. 9.

    Find the local maximum value of . You can type e.

    Show full solution

    . The signs are + on , − on , + on . The local maximum is .

  10. 10.

    Find every x-value where has a local extremum. Separate them with commas.

    Show full solution

    , so . Critical points: and (undefined). The signs are + on , − on , + on . So there's a local maximum at and a local minimum at .

  11. 11.

    At what x-value in does have a local maximum? You can type pi.

    Show full solution

    when : or . The signs are + on , − on , + on . The local maximum is at .

  12. 12.

    Suppose . Find every x-value where has a local extremum. Separate them with commas.

    Show full solution

    The factor never changes sign, so is neither. The sign of is + on , − on , + on . So there's a local maximum at and a local minimum at .
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

First Derivative Test Practice Problems

These problems follow the first derivative test lesson. Each one asks you to find a local maximum or minimum from the sign of the derivative.

The method

  1. Find and factor it.
  2. Find the critical points, where or is undefined.
  3. Test the sign of in each interval between them.
    • to − is a local max, − to + is a local min, no change is neither.

Common mistakes to watch for

  • Calling every critical point an extremum. A squared factor in usually means no sign change.
  • Missing points where is undefined, such as for .
  • Giving the x-value when the question asks for the value, or the other way around.

Frequently Asked Questions

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

Read the question. "Where" or "at what x-value" wants the location c. "Find the local maximum value" wants f(c). For x² − 8x + 3, the minimum is at x = 4 and the minimum value is −13.

How should I type exact answers?+

Fractions, pi and e are accepted, as in pi/6 or 4/e^2, and so are decimals. For more than one x-value, separate them with commas in any order.

More practice problems

Critical Points Practice Problems Mean Value Theorem Practice Problems Extreme Value Theorem Practice Problems