Concavity, inflection points and the second derivative test

Answer Key - Grade 12 Math Worksheet

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

1. 1.

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

   • A local maximum at x = 3
   • An inflection point at x = 3
   • A local minimum at x = 3
2. 2.

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

   • A local minimum at x = −1
   • An inflection point at x = −1
   • A local maximum at x = −1
3. 3.

   Kira finds that f(x) = x³ − 12x + 1 has f′(2) = 0. By the second derivative test, what happens at x = 2?

   • A local minimum at x = 2
   • An inflection point at x = 2
   • A local maximum at x = 2
4. 4.

   Noah graphs f(x) = x³ + 3x² − 9x + 2. Where is f concave up, and where concave down?

   • Up for x < −1, down for x > −1
   • Up for x > 1, down for x < 1
   • Up for x > −1, down for x < −1
5. 5.

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

   • Up for x > 2, down for x < 2
   • Up for x < 2, down for x > 2
   • Up for x < −1, down for x > −1
6. 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. 7.

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

   -11
8. 8.

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

   22
9. 9.

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

   5
10. 10.

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

    -50