1. Warm Up

2. 5.3C – Second Derivative test

3. Review One way to find local mins and maxs is to make a sign chart with the critical values. There is a theorem that can do the same thing, sometimes with less work

4. Second Derivative Test Used to find Local Extrema 1.If f’(c)=0 and f’’(c)<0 then x=c is a local max 2.If f’(c)=0 and f’’(c)>0 then x=c is a local min This test only requires us to know f’’ at c itself, which makes it easy to apply.

5. Note If f’’(c)=0 or fails to exist, we can’t use this rule and you have to go back to the first derivative sign chart to find max and mins.

6. Example

7. You try!

8. Test prep Questions:

9. Test Prep! If f’(x) = (x 2 – 4), which of the following is true? A. f has a relative minimum at x = -2 and a relative maximum at x = 2 B. f has a relative minimum at x = 2 and a relative maximum at x = -2 C. f has relative minima at x = -2 and x = 2 D. f has relative maxima at x = -2 and x = 2 E. It cannot be determined if f has any relative extrema

10. The graph of a twice-differentiable function f is shown in the figure. Which of the following is true? 1. f(1)<f'‘(1)<f‘(1) 2. f‘(1)<f(1)<f'‘(1) 3. f'‘(1)<f(1)<f‘(1) 4. f(1)<f‘(1)<f'‘(1) 5. f'‘(1)<f‘(1)<f(1)

11. Homework See syllabus for 5.3 Day 3.