Concavity and the Second Derivative Test

1. Concavity and the Second Derivative Test Calculus 3.4

2. Concavity • f is differentiable on an open interval • The graph is concave upward if f´ is increasing on the interval. • Graph lies above its tangent lines • Graph “holds water” • The graph is concave downward if f´ is decreasing on the interval. • Graph lies below its tangent lines • Graph doesn’t “hold water” Calculus 3.4

3. Finding Concavity • If f´´ > 0; the graph of f is concave up. • f´ is increasing • If f´´ < 0; the graph of f is concave down. • f´ is decreasing • If f´´ = 0, the graph of f is linear, and neither concave up nor concave down. • then f´ is constant Calculus 3.4

4. Points of inflection • The concavity of f changes • The graph of f crosses its tangent line. • May occur when f´´ is 0 or undefined. Calculus 3.4

5. Examples • Determine the points of inflection and discuss the concavity of the graph of the function. Calculus 3.4

6. The Second Derivative Test • If f´(c) = 0 and f´´(c) > 0, then f(c) is a relative minimum. • If f´(c) = 0 and f´´(c) < 0, then f(c) is a relative maximum. • If f´(c) = 0 and f´´(c) = 0, then the test fails. • Use the first derivative test Calculus 3.4

7. Examples • Determine the points of inflection and relative extrema for the function. Use the second derivative test if possible. Calculus 3.4

8. Examples • Determine the relative extrema for the function. Use the second derivative test if possible. Calculus 3.4

9. Example • Sketch the graph of a function f having the following characteristics Calculus 3.4

10. Example • The graph of f is shown. On the same set of axes, sketch the graphs of f´ and f´´. Calculus 3.4