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Second Derivative Test Practice Problems

Solve f′(x) = 0, plug each critical point into f″, and read the sign: negative is a local max, positive is a local min, and zero means the test can't decide. 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?

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    at , and . So there's a local minimum at .

  2. 2.

    has a critical point at . Find .

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    and , so . It's positive, so is a local minimum.

  3. 3.

    Find the local minimum value of .

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    at , and . , so the local minimum is .

  4. 4.

    Find the local maximum value of .

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    , so the critical points are 0 and 2. , and . The local maximum is .

  5. 5.

    Find the local maximum value of .

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    and . , so the local maximum is . At , : those are local minima.

  6. 6.

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

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    and . , so both are local minima. is a local maximum.

  7. 7.

    Find the local minimum value of for .

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    at . , so . The local minimum is .

  8. 8.

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

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    , so . , and . The local maximum is .

  9. 9.

    Find the local maximum value of .

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    at , and , so . The local maximum is .

  10. 10.

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

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    when : or . . is a local minimum, and is a local maximum.

  11. 11.

    has two critical points. At which one is the second derivative test inconclusive?

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    , so the critical points are 0 and . . (a local min), but , so the test can't decide at . The first derivative test shows it's neither.

  12. 12.

    Find the value of so that has a local minimum at .

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    . Setting gives . Check: , so , a local minimum.
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MPhil in Mathematics, Founder of Mathovia
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Second Derivative Test Practice Problems

These problems follow the second derivative test lesson. Each one asks you to classify a critical point using the sign of .

The method

  1. Find and solve .
  2. Find and plug in each critical point.
  3. is a local max, is a local min.
  4. If , the test is inconclusive: use the first derivative test.

Common mistakes to watch for

  • Mixing up the signs. Negative means concave down, so a maximum.
  • Reading as “neither”. It means the test can’t decide.
  • Giving the x-value when the question asks for the value, or the other way around.

Frequently Asked Questions

What if f″(c) = 0?+

The second derivative test is inconclusive there. Use the first derivative test: check the sign of f′ on both sides of c. For x⁴ + 4x³ at x = 0, f′ is positive on both sides, so it's neither a max nor a min.

How should I type exact answers?+

Fractions, pi and e are accepted, as in pi/3 or 1/(2e), and so are decimals. For more than one x-value, separate them with commas in any order.

More practice problems

First Derivative Test Practice Problems Critical Points Practice Problems Higher-Order Derivatives Practice Problems