Second derivatives and curve shape

Answer: 6

1. f′(x)=3x2−36x+2.
2. f′′(x)=6x−36, which changes sign at x=6.
3. The change in concavity establishes an inflection there.

Answer: Stationary point of inflection

1. f′(x)=12(x+1)2, so f′(−1)=0.
2. f′′(x)=24(x+1) changes from negative to positive at x=−1.
3. The change of concavity establishes a stationary point of inflection.

Answer: 52

1. v(t)=20t3−8t.
2. Acceleration is v′(t)=60t2−8.
3. At t=1, acceleration is 52.