1. The Shape of the Graph 3.3

2. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on I if for all points x 1 and x 2 in I, x 1 < x 2 → f(x 1 ) < f(x 2 ). 2.f decreases on I if for all points x 1 and x 2 in I, x 1 < x 2 → f(x 2 ) < f(x 1 ).

3. Corollary 3: The First Derivative Test for Increasing and Decreasing Suppose that f is continuous on [a, b] and differentiable on (a, b). If f’ > 0 at each point of (a, b) then f increases on [a, b]. If f’ < 0 at each point of (a, b) then f decreases on [a, b].

4. Example 1: Using the First Derivative Test for Increasing & Decreasing Find the critical points of f(x) = x 3 – 12x – 5 and identify the intervals on which f is increasing and decreasing.

5. First Derivative Test for Local Extrema At a critical point x = c, 1.f has a local minimum if f ’ changes from negative to positive at c. 2.f has a local maximum if f ’ changes from positive to negative at c. 3.f has no local extreme if f ’ has the same sign on both sides of c. At endpoints there is only one side to consider.

6. Definition: Concavity The graph of a differentiable function y = f(x) is 1.Concave up on an open interval I if y’ is increasing on I. 2.Concave down on an open interval I if y’ is decreasing on I.

7. Second Derivative Test for Concavity The graph of a twice-differentiable function y = f(x) is 1.Concave up on any interval where y’’ > 0. 2.Concave down on any interval where y’’ < 0.

8. Example 2: Determining Concavity Determine the concavity of y = 3 + sin x on [0, 2 π ].

9. Definition: Point of Inflection A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.

10. Theorem 5: Second Derivative Test for Local Extrema 1. If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x = c. 2. If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x = c. WARNING: This test will fail if f’’(c) = 0 or if f’’(c) does not exist. Then you must refer back to the First Derivative Test.

11. Um…ok? Read through Example 10 on page 252.