Math Problem Statement
Solution
To solve this problem, we need to analyze the behavior of the function on the open interval .
Step-by-Step Solution
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Domain Analysis: The function is defined as long as . Therefore, . The interval given is , which excludes the endpoints, so we are within the domain.
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Behavior at the Boundaries:
- As , , since .
- As , , as the denominator approaches zero from the positive side, making grow larger.
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Critical Points: Since the function is of the form , it is monotonic (increasing) on the interval . Therefore, there are no critical points within .
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Finding Absolute Maximum and Minimum:
- Absolute Minimum: The smallest value of on occurs as , where . Therefore, the absolute minimum value is .
- Absolute Maximum: Since approaches infinity as , there is no absolute maximum value (it is unbounded as approaches 8 from the left).
Conclusion:
The answers are:
- Absolute Minimum Value:
- Absolute Maximum Value: None
Since none of the answer options provided in the image matches this exactly, it seems there might be an error in the options, as they don't consider as the minimum or recognize the unbounded maximum behavior approaching infinity.
Would you like further details on this analysis?
Related Questions:
- How would the answer change if the interval included the endpoints?
- Can you find critical points for a rational function like this one?
- What does it mean when a function is unbounded on an interval?
- How do you determine the domain of a rational function?
- Why does as ?
Tip:
When dealing with rational functions, always examine the limits at the boundaries of the interval, as they often determine the behavior of maximum and minimum values.
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Rational Functions
Formulas
f(x) = 8 / (8 - x)
Theorems
None required; limit analysis at boundaries
Suitable Grade Level
Grades 11-12 or early college level
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