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

Function f(x) is shown below

At the green point the function is concave down.
At the maroonpoint 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).

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

Function f(x) is shown below

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 Concavity1. 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 ConcavityIn 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,¥).

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.