Question
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For the function:
on the interval ,
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:
  1. Evaluate the function at the endpoints of the interval.
  2. Find the critical points by taking the derivative and setting it to zero.
  3. Evaluate the function at the critical points within the interval.
  4. 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 .

Answered by UpStudy AI and reviewed by a Professional Tutor

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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 .

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