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 and concave
downward if f’ is decreasing on the interval.
What does it mean if f’ is increasing?
It’s derivative must be positive:
f’’ > 0
Concavity uses the Second derivative
Find the intervals of concavity for the function
f ( x)  2 x  3x  12 x  5
3
2
Plan of attack:
1)Find the Second derivative and set it
equal to 0, this will divide the x-axis into
intervals
2)Evaluate a test point to the left and right
of each x-value found in (1)
f ( x)  2 x  3x  12 x  5
3
2
f ' ( x)  6 x  6 x  12
2
f ' ' ( x)  12 x  6
We need the Second derivative equal
to 0
f ' ' ( x)  12 x  6  0
x = 1/2
This divides the x-axis into two intervals
(-∞, 1/2)
(1/2, ∞)
(-∞, 1/2)
(1/2, ∞)
Test a point to the left and right of 1/2
Use the Second Derivative
x = -1 and x = 1
f ' ' (1)  12(1)  6  18
f ' ' (1)  12(1)  1  6
f ' ' (1)  12(1)  6  18
f ' ' (1)  12(1)  1  6
Concave Down
_ _
(-∞, 1/2)
Concave Up
+ +
(1/2, ∞)
Since the graph changed Concavity across
the point x = 1/2
The point (1/2, f(1/2) = -3/2) on the graph is
a “point of inflection”