1. 3.4 C ONCAVITY AND THE S ECOND D ERIVATIVE T EST We will learn about: Concavity -Points of Inflection - The Second Derivative Test Review If a functions wants to switch from decreasing to increasing. or visa versa, what are its options of approach/attack! (There is only three options)

2. W HAT ARE WE DOING Last class: located intervals in which a function f increases or decreases. Today: locate intervals in which the graph of f is curving upward or curving downward (Concavity) How do we find these intervals of concavity? *lines have no concavity!

3. L ET ’ S P RACTICE S OME D RAWING AND C RITICAL T HINKING !

5. T EST FOR C ONCAVITY Q: When f´´(x) = 0 what does that tell us about f Plan of attack: Just like before!

6. E XAMPLE 1: D ETERMINING C ONCAVITY

7. E XAMPLE 2: Y OUR T URN

8. P OINTS OF I NFLECTION : When the graph switches concavity Q: If this is where the functions switches concavity then it is also where what happens? A1 derivative: A2 2 nd derivative:

9. L OCATING I NFLECTION P OINTS Do what we did before but one step further 1. Locate the x-values at which f´´(x) = 0 or f´´ is undefined (don’t stop here) 2. Use these x-values to determine test intervals 3. Test the sign of f´´(x) in each test interval 4. See if between two intervals there was a concavity switch!

10. L OOK BACK AT OUR RESULTS FROM EXAMPLE 1 & 2

11. R EMEMBER : C ONVERSE OF P OINTS OF I NFLECTION T HEOREM N OT T RUE !

12. B EFORE H OW DID WE TEST TO SEE IF AN EXTREMA WAS A M INIMA OR M AXIMA ?

13. L AST E XAMPLE : U SING 2 ND D ERIVATIVE TEST

14. 3.4 H MWR :