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Concavity and Inflection Points Practice Problems

Find f″, mark where it's zero or undefined, and test its sign in each interval: + is concave up, − is concave down. An inflection point needs a sign change at a point in the domain. Give intervals in interval notation.

  1. 1.

    At what x-value does have an inflection point?

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    is negative for and positive for . The concavity changes at .

  2. 2.

    On what interval is concave up? Use interval notation.

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    when . So is concave up on .

  3. 3.

    Find the y-coordinate of the inflection point of .

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    changes sign at . The inflection point is .

  4. 4.

    On what interval is concave down? Use interval notation.

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    when . So is concave down on .

  5. 5.

    Find every x-value where has an inflection point. Separate them with commas.

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    . The sign is + on , − on , + on . It changes at both and .

  6. 6.

    On what interval is concave down? Use interval notation.

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    , which is negative between its zeros. So is concave down on , with inflection points at and .

  7. 7.

    At what x-value does have an inflection point?

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    and . Since , the sign follows , which changes at .

  8. 8.

    Find every x-value where has an inflection point. Separate them with commas.

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    . By the quotient rule, . The sign is the sign of , which changes at .

  9. 9.

    How many inflection points does have?

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    is 0 at but positive on both sides. The sign never changes, so there are no inflection points.

  10. 10.

    On what interval is concave up? Use interval notation.

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    and . is undefined at 0, negative for and positive for . So is concave up on , with an inflection point at the origin.

  11. 11.

    Find every x-value in where has an inflection point. You can type pi. Separate them with commas.

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    when : or . is − on , + on , − on , so both are inflection points.

  12. 12.

    Find the value of so that has an inflection point at .

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    . Setting gives . is linear, so it changes sign there.
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Concavity and Inflection Points Practice Problems

These problems follow the concavity and inflection points lesson. Each one asks where a function bends up or down, or where its concavity changes.

The method

  1. Find and factor it.
  2. Find where or is undefined.
  3. Test the sign of in each interval: + is concave up, − is concave down.
  4. An inflection point is where the sign changes at a point in the domain.

Common mistakes to watch for

  • Calling every zero of an inflection point. Check that the sign changes.
  • Missing points where is undefined, such as for .
  • Using instead of . The sign of tells you increasing or decreasing, not concavity.

Frequently Asked Questions

If f″(c) = 0, is there always an inflection point at c?+

No. The sign of f″ has to change. For x⁴ + 4x, f″(0) = 0 but f″ = 12x² is positive on both sides, so there's no inflection point.

How should I type my answers?+

Use interval notation for intervals, such as (-1, oo) or (0, 3). For x-values, fractions and pi are accepted, as in 3pi/4. For more than one x-value, separate them with commas in any order.

More practice problems

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