Concavity, inflection points and the second derivative test

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   For Esme's function, f″(x) = (x − 1)(x − 4)². Where does f have an inflection point?

   • At x = 1 and at x = 4
   • Only at x = 1
   • Only at x = 4
2. 2.

   For Luz's function, f″(x) = (x − 6)(x + 1)². Where does f have an inflection point?

   • Only at x = 6
   • Only at x = −1
   • At x = 6 and at x = −1
3. 3.

   The height h(t) of Milo's bean plant has h′(t) > 0 and h″(t) < 0. What does that mean?

   • It grows, but more slowly each week
   • It is getting shorter each week
   • It grows faster and faster each week
4. 4.

   The temperature T(t) of Noah's soup has T′(t) < 0 and T″(t) > 0. What does that mean?

   • It cools faster and faster each minute
   • It is warming up each minute
   • It cools, but more slowly each minute
5. 5.

   The speed s(t) of Ren's bike coasting down a gentle hill has s′(t) > 0 and s″(t) < 0. What does that mean?

   • It speeds up by more and more each second
   • It speeds up, but by less each second
   • It is slowing down each second
6. 6.

   The water depth d(t) in Zee's draining tub has d′(t) < 0 and d″(t) > 0. What does that mean?

   • It drains faster each minute
   • The water is getting deeper each minute
   • It drains, but more slowly each minute
7. 7.

   Pia wants f(x) = 2x³ + bx² + 7x to have an inflection point at x = 4. What must b be?

   -24
8. 8.

   Esme wants f(x) = 3x³ + bx² − 4x to have an inflection point at x = −2. What must b be?

   18
9. 9.

   Jun wants f(x) = −2x³ + bx² + 9x to have an inflection point at x = 5. What must b be?

   30
10. 10.

    Kira wants f(x) = 4x³ + bx² + 6x to have an inflection point at x = 1.5. What must b be?

    -18