Lesson 27
Definitions
Examples
Second
Derivative
Test
Section 3.2: Concavity and Points of Inflection
March 24th, 2014
Lesson 27
Definitions
Examples
Second
Derivative
Test
In this lesson we will discuss two new features of the graph of a
function. The first is called concavity and is very reminiscent of
the property of a function being increasing or decreasing. The
second is called an inflection point. It bears a strong
resemblence to a critical point.
Lesson 27
Definition
Definitions
Examples
Second
Derivative
Test
If the function f (x) is differentiable on the interval a < x < b,
then the graph of f is
concave upward on a < x < b if f 0 is increasing on the
interval
concave downward on a < x < b if f 0 is decreasing on
the interval
Recall that the value of the derivative at x is the slope of the
line tangent to the graph at (x, f (x)). With this in mind, we
can determine what concavity looks like on a graph.
Lesson 27
Definitions
So f is concave up on a < x < b if the slopes of the tangent
lines are increasing, and it is concave down if the slopes of the
tangent lines are decreasing.
Examples
Second
Derivative
Test
concave up (holds water)
concave down (spills water)
We can see in the first picture that the slopes of the tangent
lines are increasing as we move from the left to the right.
Similarly on the second picture the slopes of the tangent line
decrease as we move from the left to the right.
Lesson 27
Definitions
So we have discussed a graph beiing increasing or decreasing
and concave up or concave down on an interval a < x < b.
Here are all possible combinations of these properties.
Examples
c. up, inc.
Second
Derivative
Test
c. up, dec.
c. down, inc.
c. down, dec.
Note: f 0 is increasing on a < x < b ⇒ f 00 > 0 on a < x < b
f 0 is decreasing on a < x < b ⇒ f 00 < 0 on a < x < b
Definition
An inflection point is a point (c, f (c)) around which the
graph of f changes concavity. At such a point either f 00 (c) = 0
or f 00 (c) does not exist.
Lesson 27
Example
Definitions
Examples
Second
Derivative
Test
Determine where the graph of the given function is concave
upward and concave downward. Find all inflection points.
f (x) = −2x 3 + 3x 2 + 12x − 5
Finding where f is concave up or concave down is
equivalent to finding where f 00 is positive or negative.
Thus we should find where f 00 is zero or undefined and
then make a sign chart.
f 0 (x) = −6x 2 + 6x + 12
f 00 (x) = −12x + 6 = −6(2x − 1)
Thus f 00 is zero at x = 21 , and it is defined everywhere.
Lesson 27
Definitions
−
+
Examples
1
2
Second
Derivative
Test
f 00 (0) = −6(−1) > 0,
f 00 (1) = −6(1) < 0
Therefore f (x) is concave up on −∞ < x <
concave down on 12 < x < ∞.
Since f changes concavity around x =
point ( 21 , 32 ) is an inflection point.
1
2
1
2
and it is
and f ( 12 ) = 32 , the
Lesson 27
Definitions
Examples
Second
Derivative
Test
g (t) = t(t − 2)2
g 0 (t) = (t − 2)2 + 2t(t − 2) = (t − 2)(t − 2 + 2t)
= (t − 2)(3t − 2)
g 00 (t) = (3t − 2) + (t − 2)3 = 3t − 2 + 3t − 6
= 6t − 8
= 2(3t − 4)
Thus g 00 (t) is zero at t = 43 , and it is defined everywhere.
Lesson 27
Definitions
−
+
Examples
4
3
Second
Derivative
Test
g 00 (0) = 2(−4) < 0,
Therefore g (t) is concave up on
on −∞ < t < 34 .
g 00 (2) = 2(2) > 0
4
3
< t < ∞ and concave down
Since g changes concavity around t =
point ( 43 , 16
27 ) is an inflection point.
4
3
and g ( 34 ) =
16
27 ,
the
Lesson 27
Definitions
Examples
Second
Derivative
Test
Let’s now examine the relation between concavity and relative
extrema. In the first picture below, the graph is concave up. In
the second picture, the graph is concave down. In the third and
fourth pictures, the graph changes concavity.
Note that all the following marked points are critical points.
The first picture has a relative maximum, the second has a
relative minimum, and in the final two pictures the critical
point is not a relative extremum. This gives us a new test for
relative extrema.
Lesson 27
The Second Derivative Test for Relative Extrema
Definitions
Examples
Second
Derivative
Test
Suppose f 00 (x) exists on an open interval containing x = c and
that f 0 (c) = 0.
If f 00 (c) > 0, f has a relative minimum at x = c.
If f 00 (c) < 0, f has a relative maximum at x = c.
If f 00 (c) = 0 or does not exist, the test is inconclusive.
The Second Derivative Test has the advantage of requiring less
calculation than the first derivative test. You only have to
compute the value of f 00 at the critical point rather than
computing the value of f 0 at two values near the critical point.
However, it has the disadvantage of sometimes being
inconclusive.
Lesson 27
Definitions
Example
Use the Second Derivative Test to find the relative minima and
relative maxima of the given function.
Examples
Second
Derivative
Test
f (x) = x 3 + 3x 2 + 1
First we must find the critical numbers of f .
f 0 (x) = 3x 2 + 6x = 3x(x + 2) ⇒ f 0 (x) = 0 at x = −2, 0
Since f 0 (x) is continuous everywhere, the critical numbers
of f are x = −2, 0.
f 00 (x) = 6x + 6 = 6(x + 1)
f 00 (−2) = 6(−1) < 0,
f 00 (0) = 6(1) > 0
Thus by the Second Derivative Test, (−2, f (−2)) is a
relative maximum of f and (0, f (0)) is a relative minimum.