1. Concavity and the Second Derivative Test
Determine the intervals on which the graphs of functions are concave upward or concave downward.
Find the points of inflection of the graphs of functions.
Use the Second Derivative Test to find the relative extrema of functions.
Standard 4.5a

2. Concavity – the property of curving upward or downward
Concave upward
f’ is increasing

3. f’ is decreasing
Concave downward

4. Definition of Concavity
Let f be differentiable on an open interval I. The graph of f is
Concave upward on I if f’ is increasing on the interval.
Concave downward on I if f’ is decreasing on the interval.

5. Test for Concavity
Let f be a function whose second derivative exists on an open interval I.
If f ´´(x) > 0 for all x in I, then f is concave upward on I.
If f ´´(x) < 0 for all x in I, then f is concave downward on I.

6. Determine the intervals on which the graph is concave upward or concave downward.
1. Locate the x-values at which f ´´(x) = 0 or f ´´(x) is undefined.

7. Use these x-values to determine the test intervals.
Test the signs of f ´´(x) in each test interval.
Interval
(-∞, -√3)
(-√3, √3)
(√3, ∞)
Test Values
x = -2
x = 0
x = 2
Sign of f ´´(x)
f ´´(-2) > 0
f ´´(0) < 0
f ´´(2) > 0
Conclusion
Concave upward
Concave downward

9. Defintion of Point of Inflection
If the graph of a continuous function has a tangent line at point where the concavity changes from upward to downward (or vice versa) then the point is a point of inflection.

10. Property of Points of Inflection
If (c, f(c)) is a point of inflection of the graph of f, then either f ´´(c) = 0 or f ´´(c) is undefined at c.

11. Find the points of inflection of the graph.
Possible inflection point

12. Inflection point (4, 16) (-∞, 4) (4, ∞) x = 0 x = 5 f ´´(0) < 0
Concave down
Concave up
Inflection point (4, 16)

13. It is possible for the second derivative to be zero at a point that is not a point of inflection.
* You must test to be certain that the concavity actually changes.

14. Find the points of inflection and discuss the concavity of the graph of the function.

15. Possible points of inflections: x = 0, x = 3
(0, 3)
(3, ∞)
x = 1
x = 4
f ´´(1) > 0
f ´´(4) < 0
Concave upward
Concave downward
Inflection Point:

16. If f ´(c) = 0 and f ´´(c) > 0, f (c) is a relative minimum
Concave Upward
f ´´(c) > 0 c
If f ´(c) = 0 and f ´´(c) > 0, f (c) is a relative minimum

17. If f ´(c) = 0 and f ´´(c) > 0, f (c) is a relative minimum
Concave downward
f ´´(c) < 0 c
If f ´(c) = 0 and f ´´(c) > 0, f (c) is a relative minimum

18. Second – Derivative Test
Let f ´(c) = 0 and let f ´´exist on an open interval containing c.
If f ´´(c) > 0, then f(c) is a relative minimum.
If f ´´(c) < 0, then f(c) is a relative maximum.
If f ´´(c) = 0 then the test fails. Use the First Derivative Test.

19. Find the relative extrema using the Second-Derivative Test
1. Find the critical numbers.
Critical Numbers

20. (0,0) is neither a relative max or a relative min
2. Find the second derivative.
3. Plug the critical numbers into the second derivative to determine relative extrema.
Relative minimum (-1, 2)
Test fails
Relative maximum (1, 2)
(-1, 0)
(0, 1)
f ´(-1/2) >0
f ´(1/2) >0
Increasing
(0,0) is neither a relative max or a relative min

21. (0,1) is neither a relative max or a relative min
Find the relative extrema using the Second-Derivative Test.
Test fails
Relative min (3,-26 )
(-∞, 0)
(0, 3)
f ´(-1) < 0
f ´(1) < 0
Decreasing
(0,1) is neither a relative max or a relative min