Sketch the Graph of a Continuous Function That Has Two Local Maximums but No Inflection Point
Chapter 5.5: Derivatives and the Shape of a Graph
Learning Objectives
- Explain how the sign of the first derivative affects the shape of a function's graph.
- State the first derivative test for critical points.
- Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function's graph.
- Explain the concavity test for a function over an open interval.
- Explain the relationship between a function and its first and second derivatives.
- State the second derivative test for local extrema.
Earlier in this chapter we stated that if a function has a local extremum at a point
then
must be a critical point of
However, a function is not guaranteed to have a local extremum at a critical point. For example,
has a critical point at
since
is zero at
but
does not have a local extremum at
Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
Concavity and Points of Inflection
We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function.
(Figure)(a) shows a function with a graph that curves upward. As
increases, the slope of the tangent line increases. Thus, since the derivative increases as
increases,
is an increasing function. We say this function
is concave up. (Figure)(b) shows a function
that curves downward. As
increases, the slope of the tangent line decreases. Since the derivative decreases as
increases,
is a decreasing function. We say this function
is concave down.
In general, without having the graph of a function how can we determine its concavity? By definition, a function
is concave up if
is increasing. From Corollary 3, we know that if
is a differentiable function, then
is increasing if its derivative
Therefore, a function
that is twice differentiable is concave up when
Similarly, a function
is concave down if
is decreasing. We know that a differentiable function
is decreasing if its derivative
Therefore, a twice-differentiable function
is concave down when
Applying this logic is known as the concavity test.
We conclude that we can determine the concavity of a function by looking at the second derivative of
In addition, we observe that a function
can switch concavity ((Figure)). However, a continuous function can switch concavity only at a point
if
or
is undefined. Consequently, to determine the intervals where a function
is concave up and concave down, we look for those values of
where
or
is undefined. When we have determined these points, we divide the domain of
into smaller intervals and determine the sign of
over each of these smaller intervals. If
changes sign as we pass through a point
then
changes concavity. It is important to remember that a function
may not change concavity at a point
even if
or
is undefined. If, however,
does change concavity at a point
and
is continuous at
we say the point
is an inflection point of
Figure 6. Since for
the function
is concave up over the interval
Since
for
the function
is concave down over the interval
The point
is an inflection point of
Testing for Concavity
We now summarize, in (Figure), the information that the first and second derivatives of a function provide about the graph of
and illustrate this information in (Figure).
Sign of ![]() | Sign of ![]() | Is ![]() | Concavity |
---|---|---|---|
Positive | Positive | Increasing | Concave up |
Positive | Negative | Increasing | Concave down |
Negative | Positive | Decreasing | Concave up |
Negative | Negative | Decreasing | Concave down |
Figure 8. Consider a twice-differentiable function over an open interval
If
for all
the function is increasing over
If
for all
the function is decreasing over
If
for all
the function is concave up. If
for all
the function is concave down on
The Second Derivative Test
The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.
We know that if a continuous function has a local extrema, it must occur at a critical point. However, a function need not have a local extrema at a critical point. Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Let be a twice-differentiable function such that
and
is continuous over an open interval
containing
Suppose
Since
is continuous over
for all
((Figure)). Then, by Corollary 3,
is a decreasing function over
Since
we conclude that for all
if
and
if
Therefore, by the first derivative test,
has a local maximum at
On the other hand, suppose there exists a point
such that
but
Since
is continuous over an open interval
containing
then
for all
((Figure)). Then, by Corollary
is an increasing function over
Since
we conclude that for all
if
and
if
Therefore, by the first derivative test,
has a local minimum at
Figure 9. Consider a twice-differentiable function such that
is continuous. Since
and
there is an interval
containing
such that for all
in
is increasing if
and
is decreasing if
As a result,
has a local maximum at
Since
and
there is an interval
containing
such that for all
in
is decreasing if
and
is increasing if
As a result,
has a local minimum at
Note that for case iii. when then
may have a local maximum, local minimum, or neither at
For example, the functions
and
all have critical points at
In each case, the second derivative is zero at
However, the function
has a local minimum at
whereas the function
has a local maximum at
and the function
does not have a local extremum at
Let's now look at how to use the second derivative test to determine whether has a local maximum or local minimum at a critical point
where
Using the Second Derivative Test
Use the second derivative to find the location of all local extrema for
We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as At that point, we have enough tools to provide accurate graphs of a large variety of functions.
Key Concepts
2. For the function is
both an inflection point and a local maximum/minimum?
Solution
It is not a local maximum/minimum because does not change sign
3. For the function is
an inflection point?
4. Is it possible for a point to be both an inflection point and a local extrema of a twice differentiable function?
5. Why do you need continuity for the first derivative test? Come up with an example.
6. Explain whether a concave-down function has to cross for some value of
Solution
False; for example,
7. Explain whether a polynomial of degree 2 can have an inflection point.
For the following exercises, analyze the graphs of then list all intervals where
is increasing or decreasing.
8.
9.
10.
Solution
Decreasing for increasing for
11.
12.
For the following exercises, analyze the graphs of then list all intervals where
-
is increasing and decreasing and
- the minima and maxima are located.
13.
14.
15.
16.
17.
For the following exercises, analyze the graphs of then list all inflection points and intervals
that are concave up and concave down.
18.
Solution
Concave up on all no inflection points
19.
20.
Solution
Concave up on all no inflection points
21.
22.
For the following exercises, draw a graph that satisfies the given specifications for the domain The function does not have to be continuous or differentiable.
24. over
over
for all
Solution
Answers will vary
26. There is a local maximum at local minimum at
and the graph is neither concave up nor concave down.
Solution
Answers will vary
For the following exercises, determine
- intervals where
is increasing or decreasing and
- local minima and maxima of
28. over
29.
For the following exercises, determine a. intervals where is concave up or concave down, and b. the inflection points of
30.
For the following exercises, determine
- intervals where
is increasing or decreasing,
- local minima and maxima of
- intervals where
is concave up and concave down, and
- the inflection points of
31.
32.
33.
34.
35.
36.
37.
For the following exercises, determine
- intervals where
is increasing or decreasing,
- local minima and maxima of
- intervals where
is concave up and concave down, and
- the inflection points of
Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.
38. [T] over
39. [T] over
40. [T] over
41. [T]
42. [T]
44. over
45.
46.
47.
For the following exercises, interpret the sentences in terms of
48. The population is growing more slowly. Here is the population.
Solution
49. A bike accelerates faster, but a car goes faster. Here Bike's position minus Car's position.
50. The airplane lands smoothly. Here is the plane's altitude.
Solution
51. Stock prices are at their peak. Here is the stock price.
52. The economy is picking up speed. Here is a measure of the economy, such as GDP.
Solution
For the following exercises, consider a third-degree polynomial which has the properties
Determine whether the following statements are true or false. Justify your answer.
53. for some
54. for some
Solution
True, by the Mean Value Theorem
55. There is no absolute maximum at
56. If has three roots, then it has 1 inflection point.
Solution
True, examine derivative
57. If has one inflection point, then it has three real roots.
Source: https://ecampusontario.pressbooks.pub/scccalculus1/chapter/derivatives-and-the-shape-of-a-graph/
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