Question
Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of
(Type an exact answer. Use a comma to separate answers as needed.)
B. The function
has no local minimum.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Aype function
is concave upward on the subinterval(s) (
,
).
as needed.)
B. The function
is never concave upward.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The function
is concave downate answers exact answer. Use a comma on the subinterval(s)
(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers
as needed.)
B. The function
is never concave downward.
(Type an exact answer. Use a comma to separate answers as needed.)
B. The function
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Aype function
as needed.)
B. The function
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The function
(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers
as needed.)
B. The function
Ask by Ball Matthews. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
- Local Minimum: The function
has no local minimum. - Concave Upward: The function
is concave upward on . - Concave Downward: The function
is never concave downward.
Solution
To analyze the function
, we will follow these steps:
- Find the first derivative
to determine critical points and local extrema. - Find the second derivative
to analyze concavity. - Determine intervals of concavity and local extrema.
Step 1: Find the first derivative
Using the rules of differentiation, we have:
Step 2: Find critical points
Set
:
This can be rearranged to:
Multiplying both sides by
:
Taking the square root:
Taking the natural logarithm:
Step 3: Find the second derivative
Now, we find the second derivative:
Step 4: Analyze concavity
Since both terms
and
are always positive for all
, we conclude:
This means that the function
is concave upward on the entire interval
.
Summary of Findings
- The function
has no local minimum (since it is always increasing). - The function
is concave upward on the interval . - The function
is never concave downward.
Final Answers
- Local Minimum: B. The function
has no local minimum. - Concave Upward: A. The function
is concave upward on the subinterval(s) . - Concave Downward: B. The function
is never concave downward.
Answered by UpStudy AI and reviewed by a Professional Tutor
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The Deep Dive
To analyze the function
, we first find its derivative:
. Setting
to zero leads to a critical point, confirming that there is no local minimum. The function is indeed concave upward for all real numbers because the second derivative
is always positive.
In interval notation, the function is concave upward on
and concave downward on the empty set, meaning it never curves downwards. Therefore, we conclude: there is no local minimum, it’s concave upward on
, and it’s never concave downward.