What is Concavity?
Concavity refers to the way that the function opens (up or down).

Function f(x) is shown below
Cain's_picture.GIF
At the green point the function is concave down.
At the maroon point the function is concave up.

To explain, think of a bowl when the bowl is right side up - the bowl is concave up. When the bowl is upside down, the bowl is concave down.

Test for Concavity
1. If f''(x)>0 for all x values on the interval, then the graph of f is concave upward on the interval.
2. If f''(x)<0 for all x values on the interval, then the graph of f is concave downward on the interval.

Finding Concavity
In order to find the concavity of a function first you have to find the second derivative f(x)” then find where the second derivative equals zero. This will give you the intervals on which to find test points.
Differentiate
First Derivative
Differentiate
Second Derivative
Note: because f''(x)=0 when x =1 or x =-1 and f'' is defined for all real numbers, you should use the test intervals (-¥,-1),(-1,1), and
(1,¥).



Interval
-¥<x<1
-1<x<1
1<x<¥
Test Value
x=-2
x=0
x=2
Sign of f''(x)
f''(-2)>0
f''(0)<0
f''(2)>0
Conclusion
Concave Upward
Concave Downward
Concave Upward

More information can be found in Section 3.4 of your textbook (p.190).

Another example can be found here.

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus. 8th ed. Boston and New York: Houghton Mifflin Company, 2006.