LESSON X
LESSON 15 CONCAVITY AND THE SECOND DERIVATIVE TEST
Definition Let f be a function that is differentiable at c. Then
1. the graph of f is concave upward at the point [pic] if there exists an open interval [pic] containing c such that on the interval the graph of f is above the tangent line through the point P,
2. the graph of f is concave downward at the point [pic] if there exists an open interval [pic] containing c such that on the interval the graph of f is below the tangent line through the point P.
Theorem (Test for Concavity) Suppose that a function f is differentiable on an open interval containing c and that [pic] exists. Then
1. if [pic], then the graph of f is concave upward at the point [pic],
2. if [pic], then the graph of f is concave downward at the point [pic].
Definition A point on the graph of a function at which the concavity changes is called a point of inflection or an inflection point.
Example Find the interval(s) on which the following functions are concave upward and concave downward. Also, find the point(s) of inflection ( or inflection point(s).)
1. [pic]
[pic]
[pic]
Sign of [pic]: [pic] +
(
[pic]
NOTE: Since [pic] when x is in the interval [pic], then the graph of the function f is concave upward on this interval. Since [pic] when x is in the interval [pic], then the graph of the function f is concave downward on this interval.
NOTE: Since [pic] is in the domain of the function f, then an inflection point occurs when [pic]. Now, we will calculate the y-coordinate of this point:
[pic] =
[pic] = [pic] =
[pic] = [pic] = [pic] = [pic]
Answer: Concave Upward: [pic]
Concave Downward: [pic]
Inflection Point(s): [pic]
2. [pic]
[pic]
[pic] = [pic]
NOTE: [pic] [pic] [pic] [pic]
Sign of [pic]: [pic] + [pic] +
( ( (
[pic] 0 [pic]
NOTE: Since [pic] when x is in the set [pic], then the graph of the function y is concave upward on this set. Since [pic] when x is in the set [pic], then the graph of the function y is concave downward on this set.
NOTE: Since the numbers [pic], 0, and [pic] are in the domain of the
function y, then inflection points occurs when [pic], [pic], and
[pic]. Now, we will calculate the y-coordinate of these points:
For [pic]: [pic] =
[pic] = [pic]
[pic] = [pic]
NOTE: [pic]
For [pic]: [pic] =
[pic] = [pic]
[pic] = [pic]
For [pic]: [pic]
Answer: Concave Upward: [pic]
Concave Downward: [pic]
Inflection Point(s): [pic] ; [pic]; [pic]
3. [pic]
[pic]
[pic] = [pic]
Sign of [pic]: [pic] + +
( (
[pic] 0
NOTE: Since [pic] when t is in the set [pic] and the number 0 is in the domain of the function s, then the graph of the function s is concave upward on the interval [pic]. Since [pic] when t is in the interval [pic], then the graph of the function s is concave downward on this interval.
NOTE: Since [pic] is in the domain of the function f, then an inflection point occurs when [pic]. Now, we will calculate the y-coordinate of this point:
Using a calculator, we have that [pic]
= [pic] = 2132
Not using a calculator, we have that
[pic] =
[pic] = [pic] =
[pic] = [pic] = [pic] =
[pic] = [pic] = 2132
Answer: Concave Upward: [pic]
Concave Downward: [pic]
Inflection Point(s): [pic]
4. [pic]
[pic] = [pic]
NOTE: In order to obtain the second derivative of the function y, you could use the Product Rule. However, using properties of exponents, we have that
[pic] = [pic] = [pic]
Thus, [pic] = [pic]. Using the Chain Rule
again, we have that
[pic] = [pic] =
[pic]
NOTE: [pic] [pic] [pic] and
[pic] [pic] [pic] = [pic]
Sign of [pic] : + [pic] + [pic]
( ( (
0 [pic] 4
NOTE: Since [pic] when u is in the set [pic], then
the graph of the function y is concave upward on this set. Since [pic]
when u is in the set [pic], then the graph of the function y is concave downward on this set.
NOTE: Since the numbers 0, [pic], and 4 are in the domain of the function y, then inflection points occurs when [pic], [pic], and [pic]. Now, we will calculate the y-coordinate of these points:
For [pic]: [pic]
For [pic]: [pic] =
For [pic]: [pic]
Answer: Concave Upward: [pic]
Concave Downward: [pic]
Inflection Point(s): [pic] ; [pic] ; [pic]
5. [pic]
NOTE: The domain of the function f is the interval [pic].
[pic] = [pic]
[pic]
[pic] = [pic] = [pic]
NOTE: [pic]. However, [pic] is not in the domain of the
function f .
Sign of [pic]: [pic]
(
0
NOTE: Since [pic] when x is in the interval [pic], then the graph of the function f is concave downward on this interval. Since [pic] is not positive anywhere, then the graph of the function f is not concave upward anywhere.
Answer: Concave Upward: Nowhere
Concave Downward: [pic]
Theorem (The Second Derivative Test for Local Maximums and Minimums) Let c is a critical number of a function f. If f is differentiable on an open interval containing c, then
1. if [pic], then f has a local maximum occurs at [pic]and the local maximum is [pic],
2. if [pic], then f has a local minimum occurs at [pic]and the local minimum is [pic],
3. if [pic], then the test fails and you will need to use the First Derivative Test to classify whether a local maximum, local minimum, or neither occurs at [pic].
NOTE: You would use the Second Derivative Test if it was the second derivative of the function can be calculate quickly and you didn’t want the information about when the function is increasing and decreasing. Remember, if you have the sign of the first derivative, then the First Derivative Test will give you the information about local maximums and local minimums.
Examples Find the critical numbers for the following functions. If the second derivative of the following functions can be calculated quickly, then find the local maximum(s) and local minimum(s) using the Second Derivative Test. If the second derivative can not be calculated quickly, then state this and do not find the local maximum(s) and local minimum(s).
1. [pic]
NOTE: The domain of the function f is the set of all real numbers.
[pic]
Critical Number: 3
[pic] [pic] loc max occurs at [pic]
[pic]
Answer: Local Maximum(s): [pic]
Local Minimum(s): None
2. [pic]
NOTE: The domain of the function y is the set of all real numbers such that [pic] and [pic].
We differentiated this function in Lesson 9 obtaining that
[pic]
Critical Number: [pic]
The second derivative can not be calculated quickly. It would be faster to use the First Derivative Test to classify the local maximum(s) and local minimum(s). See this work in Lesson 14.
3. [pic]
NOTE: The domain of the function g is the set of all real numbers such that [pic] and [pic].
We differentiated this function in Lesson 9 obtaining that
[pic]
Critical Numbers: 0 and [pic]
The second derivative can not be calculated quickly. It would be faster to use the First Derivative Test to classify the local maximum(s) and local minimum(s). See this work in Lesson 14.
4. [pic]
NOTE: The domain of the function y is the set of all real numbers such that [pic] and [pic].
We differentiated this function in Lesson 9 obtaining that [pic]
Critical Numbers: [pic], 5
The second derivative can not be calculated quickly. It would be faster to use the First Derivative Test to classify the local maximum(s) and local minimum(s). See this work in Lesson 14.
5. [pic]
NOTE: The domain of the function h is the set of all real numbers.
We differentiated this function in Lesson 10 obtaining that
[pic] = [pic]
Critical Number(s): [pic], [pic], 3
The second derivative can not be calculated quickly. It would be faster to use the First Derivative Test to classify the local maximum(s) and local minimum(s). See this work in Lesson 14.
6. [pic]
NOTE: The domain of the function s is the set of all real numbers.
[pic] = [pic] = [pic]
Critical Number(s): [pic], 5
[pic] = [pic]
[pic] loc max occurs at [pic]
[pic] loc min occurs at [pic]
[pic] = [pic] = [pic]
[pic]
Answer: Local Maximum(s): [pic]
Local Minimum(s): [pic]
7. [pic]
NOTE: The domain of the function y is the set of all real numbers.
[pic] = [pic]
Critical Numbers: [pic], 0
[pic] = [pic]
[pic]: [pic] loc min occurs at [pic]
[pic]: [pic] the test fails
Using the First Derivative Test, we have that
Inc Inc
Sign of [pic]: X + +
( (
[pic] 0
Thus, neither a local maximum nor a local minimum occurs at [pic] since the function y is increasing on the left-hand side of [pic] and it is still increasing on the right-hand side of [pic].
NOTE: We do not need the sign of [pic] on the interval [pic] in order to classify what is happening at the critical number of [pic].
Using a calculator: [pic] or
[pic] = [pic] = [pic] =
[pic] = [pic] = [pic] = [pic]
Answer: Local Maximum(s): None
Local Minimum(s): [pic]
8. [pic]
NOTE: The domain of the function g is the set of all real numbers such that [pic].
[pic] = [pic] = [pic]
Critical Numbers: [pic]
[pic] [pic]
[pic] loc max occurs at [pic]
Since [pic], then [pic] =
[pic] = [pic] = [pic] = [pic]
Answer: Local Maximum(s): [pic]
Local Minimum(s): None
9. [pic]
NOTE: The domain of the function f is the interval [pic].
[pic] = [pic]
[pic] = [pic] = [pic]
Critical Number(s): 3
[pic] [pic] =
[pic] = [pic]
[pic] loc min occurs at [pic]
Since [pic], then [pic]
Answer: Local Maximum(s): None
Local Minimum(s): [pic]
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