Finding Concavity and Points of Inflection

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To determining concavity of a graph is similar to the method of finding increasing/decreasing intervals of a graph. Note that concavity has nothing to do with teeth, but Calculus is another word for tartar.

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 on I if f ' is decreasing on the interval. In other words, if the graph arcs like a U then it is concave upwards. If it arcs like the golden arches of McDonald's then it arcs downward.

Concavity Test Theorem
Let f be a function whose second derivative exists on an open interval I.
If f ''(x) > 0 for all x in I, then the graph of f is concave upward.
If f ''(x) < 0 for all x in I, then the graph of f is concave downward.
If f ''(x) = 0 for all x in I, then the graph of f is a line, neither concave upward or downward.

Points of Inflection
If (c, f(c) is a point of inflection on the graph of f, then either f ''(c) = 0 or is undefined at x = c. Points of Inflection are the same as critical points except they use the second derivative of f.

Finding Concavity

1.	Locate the points of inflection and use these numbers to determine test intervals.
2.	Determine the sign of f ''(x) at one value in each of the test intervals.
3.	Use the Concavity Test Theorem to decide whether f is concave upward or downward on each interval.

Example:

Determine the Concavity of the graph of f(x) = x⁴ - 4x³.

Locate the points of inflection and use them as test intervals
f '(x) = 4x³ - 12x²
f ''(x) = 12x² - 24x
12x(x - 2) = 0
x = 0, 2

see table below for test intervals

Determine the sign of f ''(x)
see table below

Use the Concavity Test Theorem
see table below

Interval	-infinity < x < 0	0 < x < 2	2 < x < infinity
Test Value	x = -1	x = 1	x = 3
Sign of f ''(x)	f ''(-1) = 36 > 0	f ''(1) = -12 < 0	f ''(3) = 36 > 0
Use Theorem	Concave upward	Concave downward	Concave upward