3.3: The Second Derivative and Curve Sketching

1. 3.3: The Second Derivative and Curve Sketching The Second derivative test

2. Concavity Concavity determines whether the graph is curving upward (concave up) or downward (concave down). If f ′′ > 0, then f is concave up at that point. If f ′′ < 0, then f is concave down at that point.

3. Example 1: Determine when the following functions are concave up or down A. is always positive, so f(x) is always concave up B. , Concave up: Concave down:

4. Points of Inflection A point of inflection is where the graph changes concavity. A point of inflection occurs when f ′′ = 0 or f ′′ does not exist. However, f ′′ mustchangesign for it to be a point of inflection. Points of inflection also occur where f ′ is a max or a min.

5. Points of Inflection Example 2: Find all points of inflection for the function

6. Example 3 Find the points of inflection for of the following function, and determine the intervals of concavity. Concave Up: Concave Down:

7. Second Derivative Test for Local Extrema • If f ′(c) = 0 and f ′′(c) < 0, then fhas a local max at x = c. • If f ′(c) = 0 and f ′′(c) > 0, then f has a local min at x = c. • If f ′(c) = 0 and f ′′(c) = 0 or DNE, then must use First Derivative Test to determine local extrema.

8. Example 4 Find the local extreme values of Rel. Min: (2,-21) Rel. Max: (2,11)

9. Example 5 Use the second derivative test to find and classify the relative extrema for the following function Relative Max at (0,0) Relative Min at (1,-13)