Math Problem Statement

Find the absolute maximum value and the absolute minimum value, if any, of the function f(x) = 8 / (8 - x) on (0, 8).

Solution

To solve this problem, we need to analyze the behavior of the function f(x)=88xf(x) = \frac{8}{8 - x} on the open interval (0,8)(0, 8).

Step-by-Step Solution

  1. Domain Analysis: The function f(x)=88xf(x) = \frac{8}{8 - x} is defined as long as 8x08 - x \neq 0. Therefore, x8x \neq 8. The interval given is (0,8)(0, 8), which excludes the endpoints, so we are within the domain.

  2. Behavior at the Boundaries:

    • As x0+x \to 0^+, f(x)1f(x) \to 1, since f(0)=880=1f(0) = \frac{8}{8 - 0} = 1.
    • As x8x \to 8^-, f(x)+f(x) \to +\infty, as the denominator 8x8 - x approaches zero from the positive side, making f(x)f(x) grow larger.
  3. Critical Points: Since the function f(x)f(x) is of the form 88x\frac{8}{8 - x}, it is monotonic (increasing) on the interval (0,8)(0, 8). Therefore, there are no critical points within (0,8)(0, 8).

  4. Finding Absolute Maximum and Minimum:

    • Absolute Minimum: The smallest value of f(x)f(x) on (0,8)(0, 8) occurs as x0+x \to 0^+, where f(x)1f(x) \to 1. Therefore, the absolute minimum value is 11.
    • Absolute Maximum: Since f(x)f(x) approaches infinity as x8x \to 8^-, there is no absolute maximum value (it is unbounded as xx approaches 8 from the left).

Conclusion:

The answers are:

  • Absolute Minimum Value: 11
  • 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 f(x)1f(x) \to 1 as the minimum or recognize the unbounded maximum behavior approaching infinity.

Would you like further details on this analysis?

Related Questions:

  1. How would the answer change if the interval included the endpoints?
  2. Can you find critical points for a rational function like this one?
  3. What does it mean when a function is unbounded on an interval?
  4. How do you determine the domain of a rational function?
  5. Why does f(x)+f(x) \to +\infty as x8x \to 8^-?

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