1. 4.4 Concavity and the Second Derivative Test
Rita Korsunsky

6. Concavity +

7. __

8. A Point of Inflection occurs at the point where the concavity changes.+ PI PI PI

9. Example 1 + _ +
Concave down
Concave up

10. Solution: Example 2 f(x) concave up f’’(x)= [f’(x)]’ >0
It means f’(x) is increasing.
f’(x) is increasing on (2,4)
f(x) is concave up on (2,4)
y=f’(x)
The graph of f’(x) on the interval [-3,4] is shown above. On what intervals is the graph of f(x) concave up?

11. 2nd Derivative Test for Local Max and Min+

12. Example 3 - - The critical points are x = -2 , 0
Use second derivative test
- -
max
Therefore, no conclusion
Since the 2nd derivative test is not applicable when x = 0, we use the 1st derivative test. - +

13. For concavity, first find the points of inflection.
EXample 3 continues
For concavity, first find the points of inflection. 1 - - +
Concave down
Concave down
Concave up +

14. Constructing first and second derivative graphs
Construct f’ if f is given
Practice graphing
Quiz on derivatives and graphing

15. 4.4 #32
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