Math Problem Statement
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Solution
To find the absolute minimum and maximum values of the function on the interval , we need to follow these steps:
Step 1: Compute the Derivative
First, let's find the derivative to identify the critical points within the interval.
The derivative is:
Step 2: Set the Derivative to Zero
We set to find critical points.
This can be rearranged to:
Dividing both sides by 8:
Multiply both sides by :
Taking the ninth root of both sides gives:
However, is not within the interval , so we discard this solution.
Step 3: Evaluate at the Endpoints
Since there are no critical points in the interval, we evaluate the function at the endpoints and .
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At : Calculating each part:
So,
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At : Calculating each part:
So,
Conclusion
- The absolute minimum value of on the interval is at .
- The absolute maximum value of on the interval is at .
Would you like more details on this process or any specific step?
Follow-up Questions
- What is the importance of checking endpoints when finding absolute extrema on a closed interval?
- How would the process differ if the interval were open (e.g., )?
- What would happen if a critical point did exist within the interval?
- How do derivatives help identify local maxima and minima in general?
- Can you explain why was not considered a valid solution in this interval?
Tip: When solving for extrema on a closed interval, always check both critical points and endpoints.
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Math Problem Analysis
Mathematical Concepts
Calculus
Optimization
Critical Points
Absolute Maximum and Minimum
Formulas
f(x) = x^8 + 8/x
f'(x) = 8x^7 - 8/x^2
Theorems
Extreme Value Theorem
Suitable Grade Level
College Calculus