https://file.aiquickdraw.com/mathbot/file/1730002049652twyk472q.jpg

To analyze this problem, let’s break down each part.

Given the function:
\[
f(x) = \frac{3 - x}{3(x - 18)^4}
\]

### Part (a): Domain of \( f \)
The domain of \( f(x) \) is determined by points where the function is undefined. Specifically, the denominator \( 3(x - 18)^4 \) must not be zero. This happens when \( x = 18 \), which would make the denominator zero and the function undefined. Therefore, the domain is:
\[
(-\infty, 18) \cup (18, \infty)
\]
which you’ve already entered correctly.

### Part (b): Intervals of Increase and Decrease
To find where \( f(x) \) is increasing or decreasing, we’ll calculate \( f'(x) \), the derivative of \( f(x) \), and analyze its sign over the intervals defined by \( x = 18 \).

1. **Find \( f'(x) \):**
   Use the quotient rule for derivatives, since \( f(x) \) is a quotient of \( 3 - x \) and \( 3(x - 18)^4 \).

   Let’s set:
   - \( u = 3 - x \)
   - \( v = 3(x - 18)^4 \)

   Then, \( f(x) = \frac{u}{v} \), and by the quotient rule:
   \[
   f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}
   \]
   where \( u' = -1 \) and \( v' \) can be found by differentiating \( 3(x - 18)^4 \).

2. **Evaluate the sign of \( f'(x) \) around \( x = 18 \):**
   Substitute values from each interval \( (-\infty, 18) \) and \( (18, \infty) \) into \( f'(x) \) to determine if the derivative is positive or negative in those intervals.

This analysis will reveal where \( f(x) \) is increasing or decreasing.

Would you like me to go into further detail on finding \( f'(x) \) and evaluating it at test points? 

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### Follow-Up Questions
1. How would the behavior of \( f(x) \) change if the exponent in the denominator were different?
2. What impact does the discontinuity at \( x = 18 \) have on the overall graph of \( f(x) \)?
3. Could there be any asymptotes associated with \( f(x) \) based on the domain?
4. How can we interpret the behavior of \( f(x) \) at infinity and near \( x = 18 \)?
5. How does the power of \( (x - 18)^4 \) affect the intervals of increase and decrease?

### Tip
When analyzing functions with denominators, always start by checking for values that make the denominator zero to identify critical points and discontinuities.

Assignment 2: Problem 11. a) Give the domain of f (in interval notation). b) Determine the intervals on which f is increasing and decreasing. The function given is f(x) = (3 - x) / [3(x - 18)^4].

This problem involves finding the domain of f(x) = (3 - x) / [3(x - 18)^4] and determining the intervals where the function is increasing or decreasing. It includes applying the quotient rule for derivatives and analyzing the function’s behavior across intervals.

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