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Grade 12 · Math · Calculus

12th grade Concavity, inflection points and the second derivative test worksheet

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Concavity, inflection points and the second derivative test

Grade 12 Math Worksheet

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Circle the best answer, or write your answer on the line.

  1. Esme's function has f′(3) = 0 and f″(3) = 5. What happens at x = 3?

    • An inflection point at x = 3
    • A local maximum at x = 3
    • A local minimum at x = 3

  2. Jun's function has f′(−1) = 0 and f″(−1) = −4. What happens at x = −1?

    • An inflection point at x = −1
    • A local maximum at x = −1
    • A local minimum at x = −1

  3. Luz finds that g(x) = 27x − x³ has g′(3) = 0. By the second derivative test, what happens at x = 3?

    • A local minimum at x = 3
    • A local maximum at x = 3
    • An inflection point at x = 3

  4. Milo graphs f(x) = x³ − 6x² + 9x + 1. Where is f concave up, and where concave down?

    • Up for x < 2, down for x > 2
    • Up for x > 3, down for x < 3
    • Up for x > 2, down for x < 2

  5. Ren graphs f(x) = −x³ + 6x² + 15x. Where is f concave up, and where concave down?

    • Up for x < −1, down for x > −1
    • Up for x > 2, down for x < 2
    • Up for x < 2, down for x > 2

  6. Zee graphs f(x) = 2x³ − 3x² − 12x. Where is f concave up, and where concave down?

    • Up for x < 1/2, down for x > 1/2
    • Up for x > 2, down for x < 2
    • Up for x > 1/2, down for x < 1/2

  7. Pia studies f(x) = x³ − 12x + 5. It has two critical points. What is the local minimum value of f?

  8. Esme studies f(x) = x³ − 48x + 150. It has two critical points. What is the local minimum value of f?

  9. Jun studies f(x) = x³ − 3x + 7. It has two critical points. What is the local minimum value of f?

  10. Kira studies f(x) = x³ − 75x + 200. It has two critical points. What is the local minimum value of f?

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12th grade Concavity, inflection points and the second derivative test worksheet – printable
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About this worksheet

Concavity, inflection points and the second derivative test. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can find f″ and use its sign to say where a graph is concave up or down.
  • I can find inflection points, including when a zero of f″ is not one.
  • I can use the second derivative test to call a critical point a local max or min.
  • I can read f′ and f″ together in a story about growth or cooling.

Standard: none (beyond CCSS)

Words to know

concave up: Bending like a cup; f″ > 0.

concave down: Bending like a cap; f″ < 0.

inflection point: Where the bend changes, because f″ changes sign.

second derivative test: At f′ = 0: f″ > 0 is a valley, f″ < 0 a peak.

Common mistakes

Found f′ instead of f″. Take the derivative once more.

Dropped the power that comes down when taking a derivative, so bx² gave b instead of 2b.

Solved f″ = 0 but lost the minus sign. Put the answer back into f″ to check.

Questions parents ask

What is the most common mistake with concavity, inflection points and the second derivative test?

Found f′ instead of f″. Take the derivative once more.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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