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Roberto’s Notes on Differential Calculus
Chapter 8 : Graphical analysis Section 3
Second derivative analysis
What you need to know already: What you can learn here:
How to perform a first derivative analysis and
interpret its result as information about the
up/down pattern of the graph.
How to analyze the second derivative and
interpret the results as information about the
concavity of the graph.
The first derivative provides information about the increase/decrease
pattern of a function.
The second derivative is the derivative of the first derivative: don’t get
confused by the words!
These two facts, together, imply the following.
Knot on your finger
The second derivative provides information on the
increase/decrease pattern of the first derivative, that
is, of the slope.
But what does that mean for the original function? Knowing how the slope is
changing tells us how the graph is curving. Here is how.
Technical fact
If f ( )^ x 0 then the slope of the
curve is increasing, so that the curve
is concave up and will present a
shape similar to a section of the
graph shown here.
Proof
Not much to prove, since being concave up actually means having an
increasing slope! Just notice that as you follow the graph shown from left to
right, the slope does indeed get bigger, whether the function is decreasing
(the negative slope becomes less negative) or increasing (the positive slope
becomes more so).
And of course the opposite situation is symmetrically true.
Technical fact
If (^) f ( ) (^) x 0 then the slope of the
curve is decreasing, so that the curve
is concave down and will present a
shape similar to a section of the
graph shown here.
These two facts take care of the points where f ( )^ x 0 or f ( )^ x 0 , so the
only remaining cases relate to where the second derivative is 0 or undefined. At
these points the concavity may change and we need one last definition.
Definition
A point c , f c ( )on the graph of a function is an
inflection point if:
the function is continuous at x c and
the second derivative changes sign as x goes
from the left to the right of c.
I remember that in high school they told us that an inflection point was one
where the second derivative was 0: does it have to be?
That is an interesting point on which you will probably see different opinions.
Some authors require that the second derivative be continuous at an inflection point,
thus limiting the definition to the case where f ^ x 0. They do that to avoid
having to call inflection point a point where the curve takes a sharp change. I
understand the feeling, which is based on a technical issue. But I prefer to focus on
the purpose : why do we want to identify a point as being an inflection point? The
most practical answer is that the concavity changes there, so I stay with the more
generous definition that allows for the second derivative to be undefined there. It is
the point’s perspective on the concavity that is of interest in an inflection point.
We are soon going to see some examples, but let me conclude the theoretical
part first, by summarizing the strategy for a second derivative analysis. If you get
feelings of déjà vu, it’s because you have seen the same strategy not long ago!
Strategy for performing a
Second derivative analysis
To determine the pattern of concavity in the graph of
y f ( ) x and to identify its inflection points :
1. Use standard algebraic methods to find the cut
points of (^) f x .
2. Place these values on a number line , as in the
strategy to solve an inequality.
3. Test each resulting interval, to see if the
function is concave up or down there, and
indicate this information on the number line.
4. Follow the concavity pattern and use any other
feature of the function to classify each cut point
as an inflection point or another feature.
The second derivative analysis also allows us to identify a special feature of a
graph that may not be identifiable by using only the first derivative analysis.
Definition
A cusp occurs in the graph of a function at a point
where:
The function is continuous,
The tangent line is vertical,
The concavity does not change.
Example:
3 2
f x x
The first and second derivatives are as follows:
2 1 4 3 2 3 2 3
3 9
f x x f x x f x x
The negative exponents of the two derivatives tell us that both become
infinite at x 0. However, the function is continuous there and the
derivative is positive on both sides of it. The graph clearly shows the “cuspy”
nature of the point.
Are inflection points and cusps the only features that can occur at a critical
value for the second derivative?
The straight answer to your question is NO, but be careful! In wording your
question you included a common error and seemed to ignore something we already
saw in the first derivative analysis. First of all…
Warning bells
A critical value is a feature of a function related to its
first derivative only. The cut points of the second
derivative do not have a special name.
Oops, sorry! And what did I forget?
Knot on your finger
A cut point of the second derivative may correspond
to a discontinuity. In particular it may correspond to
a vertical asymptote.
X
Y
Oh, right! Is that it then?
As I said, the answer to your questions is no, as there are other possible
features. One in particular is worth noting.
Technical fact
A cut point of the second derivative may also
correspond to a point where the function is
continuous, but has a tangent line coming from the
left side that is different from the one obtained
coming from the right side.
Such points are sometimes referred to as sharp points
and you will see them here only in connection to
piecewise functions, although they may occur in
other, more complex functions as well.
Example:
2
cos( 2) 1 2
x x if x
f x
x if x
^ ^
The graph of this function is shown here.
You can see that at the point 2, 2there is a sharp change of direction. That
is why we call this a sharp point. Notice that in this case 2, 2is also an
inflection point, but this may not be the case in other situations.
Also, by computing the first and second derivative, you can check (do it!) that
the function has different slopes if we look at the left and right side, that is:
0 0
lim lim h h
f x h f x f x h f x
h h ^
Hence we can think of there being two different tangent lines at this point, as
shown here. Of course the function has no unique tangent line there, as the
tangent line must be unique to exist, but it is possible to refer to a left and
right tangent.
I will conclude this section by mentioning a popular, but rather ineffective use
of the second derivative.
Technical fact:
The Second Derivative Test
If x c is a critical value of y f (^) x , then:
If f c 0 , then (^) c f , (^) c is a relative
minimum for f x .
If f ^ c 0 , then (^) c f , c is a relative
maximum for f (^) x .
Memory questions:
- What does a positive second derivative imply for the graph of a function?
- What does a negative second derivative imply for the graph of a function?
- What identifies an inflection point?
- Which cut points for the second derivative are inflection points for the graph?
- What is the purpose of the Second Derivative Test?
Computation questions:
Perform a full second derivative analysis on the functions presented in questions 1-19.
5 3 y x 10 x
4 2 y x 4 x
2
x 1 y x x
3
y x x
.
-
3
3
x f x x
2
x f x x
.
-
2/3 2/ f x x 1 x 1
-
2 1 3 3 f x x 2 x
- 2
x f x
x
- 2
x f x
x
- 3
1 2 x f x x
-
2 x 1 f x
x
- (^)
3 3 f x x x 1
-
2 3 f x x x
-
x^2 e f x x
x^2 x^3
y e
-
2 y ln x 2 x 2
2 y x ln x
2
y sin x 2sin x , 0 x 2
- A function has a derivative whose graph is shown below. From it conduct a first and second derivative analysis of the original function and briefly describe your method.
Consider each tick mark as one unit.
Theory questions:
- Can an inflection point occur at a sharp corner (cusp)?
- Can an inflection point occur where the first derivative is undefined?
- Mention one function that has infinitely many inflection points.
- If at a certain point y” is undefined, but it is positive on both side of the point,
what two graphical feature can occur?
- Can the second derivative on either side of a sharp corner be of the same sign?
- Is every point for which y ”=0 an inflection point?
- Can the second derivative equal 0 at a point with a vertical tangent line?
- Does the Second Derivative Test tell us that the second derivative at a minimum
is positive?
- Can a function have a local maximum at an inflection point?
- What graphical feature occurs at a point where the function is continuous and its
second derivative is undefined, but keeps the same sign?
- Which algebraic method is the basis for both the first and second derivative
analysis?
- What graphical feature occurs at a point where the a function is continuous and
its second derivative changes from positive to negative?
- What two graphical features may occur at a critical value that does not generate
a maximum or a minimum?
- Which cut points of the second derivative are critical values?
- Is it possible for a function to have a maximum at a point where the second
derivative does not exist?
- How good is the linear approximation of a twice differentiable function at an
inflection point?
- If the derivative of a function is decreasing, how is the concavity of the function
itself?
- If a function is continuous, but is not differentiable at x =0, does t mean that it
has a sharp corner there?
X
Y
