First Derivative Test, Concavity, Points of Inflection

1. First Derivative Test, Concavity,Points of Inflection Section 4.3a

2. Do Now Writing: True or False – A critical point of a function always signifies an extreme value of the function. Explain. FALSE!!! – Counterexample???

3. As we’ve seen, whether or not a critical point signifies an extreme value depends on the sign of the derivative in the immediate vicinity of the critical point: Abs. Max. Local Max. No Extreme No Extreme Local Min.

4. First Derivative Test for Local Extrema The following test applies to a continuous function . At a critical point c: 1. If changes sign from positive to negative at c then has a local maximum value at c. Local Max. Local Max. c c undefined

5. First Derivative Test for Local Extrema The following test applies to a continuous function . At a critical point c: 2. If changes sign from negative to positive at c then has a local minimum value at c. Local Min. Local Min. c c undefined

6. First Derivative Test for Local Extrema The following test applies to a continuous function . At a critical point c: 3. If does not change sign at c, then has no local extreme values at c. No Extreme No Extreme c c undefined

7. First Derivative Test for Local Extrema The following test applies to a continuous function . At a left endpoint a: If for x > a, then f has a local maximum (minimum) value at a. Local Max. Local Min. a a

8. First Derivative Test for Local Extrema The following test applies to a continuous function . At a right endpoint b: If for x < b, then f has a local minimum (maximum) value at b. Local Max. Local Min. b b

9. To use the first derivative test: • Find the first derivative and any critical • points • Partition the x-axis into intervals using • the critical points • Determine the sign of the derivative in • each interval, and then use the test to • determine the behavior of the function

10. Use the first derivative test on the given function Critical point:x = 0 (derivative undefined) Intervals x < 0 x > 0 Sign of + + Behavior of Increasing Increasing Can we support these answers with a graph??? Increasing on No Extrema

11. Use the first derivative test on the given function Critical points: x = 2, –2 (derivative zero) Intervals x < –2 –2 < x < 2 x > 2 Sign of + – + Behavior of Increasing Decreasing Increasing Local Max of 11 at x = –2, Local Min of –21 at x = 2 Increasing on and Decreasing on

12. The cubing function is always increasing, and never decreasing… But that doesn’t tell the entirestory about its graph… increases Where on the graph of this function is the slope increasing where is it decreasing? decreases This leads to our definition of concavity…

13. Definition: Concavity The graph of a differentiable function is (a)Concave up on an open interval I if is increasing on I. (b) Concave down on an open interval I if is decreasing on I. Concavity Test The graph of a twice-differentiable function is (a) Concave up on any interval where . (b) Concave down on any interval where . A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.

14. Let’s work through #8 on p.204 CP: IP: Intervals x < 0 0 < x < 1 1 < x < 2 2 < x – – + + Sign of + + – – Sign of Dec Conc up Inc Conc up Inc Conc down Dec Conc down Behavior of Establish some graphical support!!!

15. Let’s work through #8 on p.204 (a) Increasing on (b) Decreasing on (c) Concave up on (d) Concave up on (e) Local maximum of 5 at Local minimum of –3 at (f) Inflection point