Question
For the function:
on the interval
,
The absolute min occurs at
The absolute max occurs at
The absolute min occurs at
The absolute max occurs at
Ask by Weber Ayala. in the United States
Mar 07,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The absolute maximum occurs at
.
Solution
To find the absolute maximum of the function
on the interval
, we need to follow these steps:
- Evaluate the function at the endpoints of the interval.
- Find the critical points by taking the derivative and setting it to zero.
- Evaluate the function at the critical points within the interval.
- Compare the values obtained from the endpoints and critical points to determine the absolute maximum.
Step 1: Evaluate at the Endpoints
First, we evaluate
at the endpoints
and
.
-
For
: -
For
: To calculate: We can approximate, so: Thus,
Step 2: Find Critical Points
Next, we find the derivative of
:
Setting the derivative to zero to find critical points:
Step 3: Evaluate at the Critical Point
Now we evaluate
at the critical point
:
Step 4: Compare Values
Now we compare the values:
The maximum value occurs at
with
.
Conclusion
The absolute maximum occurs at
.
Thus, the final answer is:
The absolute max occurs at .
The absolute max occurs at
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Bonus Knowledge
To find the absolute maximum of the function
on the interval
, we need to evaluate
at the critical points and the endpoints.
The derivative,
, can help us find critical points. Setting
leads to
, so
, which gives
.
Evaluating
at the endpoints and the critical point:
- At
: - At
: - At
:
Comparing these values:
The absolute maximum occurs at
.
So, the final statement is:
The absolute max occurs at .
The absolute max occurs at