Concavity, inflection points and the second derivative test

Answer Key - Grade 12 Math Worksheet - Building Confidence

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

1. 1.

   Esme is checking how the graph of f(x) = 2x³ + 3x² − 4x + 1 bends. What is f″(1)?

   18
2. 2.

   Jun studies f(x) = 3x³ − 2x² + 5x − 7. What is f″(2)?

   32
3. 3.

   Kira studies f(x) = −2x³ + 6x² + 3x + 4. What is f″(3)?

   -24
4. 4.

   Luz is checking how the graph of f(x) = 4x³ − 5x² − 2x + 8 bends. What is f″(−1)?

   -34
5. 5.

   f″(6) = −8 for Milo's function f. What does that say about f near x = 6?

   • f is decreasing as it passes x = 6
   • f is concave down near x = 6
   • f is concave up near x = 6
6. 6.

   Noah's function has f″(x) = 6x − 12. What does f″ say about f at x = 4?

   • f is concave up near x = 4
   • f is increasing as it passes x = 4
   • f is concave down near x = 4
7. 7.

   f″(−2) = 5 for Ren's function f. What does that say about f near x = −2?

   • f is concave up near x = −2
   • f is concave down near x = −2
   • f is increasing as it passes x = −2
8. 8.

   Zee's function has f″(x) = 3x − 15. What does f″ say about f at x = 1?

   • f is concave down near x = 1
   • f is decreasing as it passes x = 1
   • f is concave up near x = 1
9. 9.

   Pia graphs f(x) = 2x³ − 30x² + 7x + 1. At what x does f have an inflection point?

   5
10. 10.

    Esme graphs f(x) = 2x³ + 6x² − 5x + 4. At what x does f have an inflection point?

    -1