f(x) = 
9/
x
,    1 < x < 3
absolute maximum value	
  

absolute minimum value	
  

local maximum value(s)	
  

local minimum value(s)	
  


The function you provided is:

\[
f(x) = \frac{9}{x}, \quad \text{with domain } 1 < x < 3.
\]

### Steps to find the absolute and local extrema:

1. **Find the derivative** of \( f(x) \) to locate critical points (possible local extrema).
   
   \[
   f'(x) = -\frac{9}{x^2}.
   \]
   
   The derivative is negative for all \( x \) in the interval \( (1, 3) \), indicating the function is decreasing on this interval.

2. **Check the critical points** by solving \( f'(x) = 0 \). However, since \( f'(x) \) is never 0, there are no critical points where the slope changes sign, meaning no local extrema inside the interval \( (1, 3) \).

3. **Check the boundary points** of the domain for absolute extrema:
   - At \( x = 1 \):
     \[
     f(1) = \frac{9}{1} = 9.
     \]
   - At \( x = 3 \):
     \[
     f(3) = \frac{9}{3} = 3.
     \]

4. **Conclusion**:
   - The **absolute maximum** occurs at \( x = 1 \), with \( f(1) = 9 \).
   - The **absolute minimum** occurs at \( x = 3 \), with \( f(3) = 3 \).
   - There are **no local maxima** or **local minima** within the interval \( (1, 3) \), as the function is strictly decreasing.

### Summary:
- **Absolute maximum value**: \( 9 \) at \( x = 1 \)
- **Absolute minimum value**: \( 3 \) at \( x = 3 \)
- **Local maximum value(s)**: None
- **Local minimum value(s)**: None

Would you like further details or have any questions?

### Related questions:
1. What happens if the interval is extended beyond \( x = 3 \)?
2. How would this function behave if the domain was \( x > 0 \)?
3. Can we generalize this result for any function of the form \( f(x) = \frac{c}{x} \)?
4. What would the critical points look like if \( f'(x) = 0 \) had real solutions?
5. How does the behavior of the function change when \( x \to \infty \)?

**Tip**: When working with rational functions like \( f(x) = \frac{9}{x} \), pay attention to critical points and the boundaries of the domain to identify extrema!

f(x) = 9/x, 1 < x < 3. Find the absolute maximum value, absolute minimum value, local maximum value(s), and local minimum value(s).

Explore how to find the absolute maximum and minimum values, as well as local extrema for the rational function f(x) = 9/x over the interval 1 < x < 3. Using calculus techniques like finding the derivative and analyzing boundary points, this problem demonstrates key concepts in identifying extrema for decreasing functions. Ideal for students in Grades 11-12 studying pre-calculus or calculus.

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