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To solve this problem, we need to find the absolute maximum and minimum values of the function \( f(x) = 3x + 3 \) over two different intervals, specifically \( [0, 3] \) and \( [-4, 5] \).

Since \( f(x) = 3x + 3 \) is a linear function with a constant slope of 3, it has no turning points or critical points within the intervals. Thus, the maximum and minimum values will occur at the endpoints of each interval.

### Part (A): Interval \( [0, 3] \)

1. **Evaluate \( f(x) \) at the endpoints of the interval**:
   - At \( x = 0 \): \( f(0) = 3(0) + 3 = 3 \).
   - At \( x = 3 \): \( f(3) = 3(3) + 3 = 9 + 3 = 12 \).

2. **Determine the absolute maximum and minimum values**:
   - The **absolute minimum value** on \( [0, 3] \) is 3, occurring at \( x = 0 \).
   - The **absolute maximum value** on \( [0, 3] \) is 12, occurring at \( x = 3 \).

So, for interval \( [0, 3] \):
- Absolute Maximum Value: **12 at \( x = 3 \)**
- Absolute Minimum Value: **3 at \( x = 0 \)**

### Part (B): Interval \( [-4, 5] \)

1. **Evaluate \( f(x) \) at the endpoints of the interval**:
   - At \( x = -4 \): \( f(-4) = 3(-4) + 3 = -12 + 3 = -9 \).
   - At \( x = 5 \): \( f(5) = 3(5) + 3 = 15 + 3 = 18 \).

2. **Determine the absolute maximum and minimum values**:
   - The **absolute minimum value** on \( [-4, 5] \) is -9, occurring at \( x = -4 \).
   - The **absolute maximum value** on \( [-4, 5] \) is 18, occurring at \( x = 5 \).

So, for interval \( [-4, 5] \):
- Absolute Maximum Value: **18 at \( x = 5 \)**
- Absolute Minimum Value: **-9 at \( x = -4 \)**

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### Final Answer Summary:
- **(A) Interval [0, 3]:**
  - Absolute Maximum Value: 12 at \( x = 3 \)
  - Absolute Minimum Value: 3 at \( x = 0 \)

- **(B) Interval [-4, 5]:**
  - Absolute Maximum Value: 18 at \( x = 5 \)
  - Absolute Minimum Value: -9 at \( x = -4 \)

Would you like further details on this solution?

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### Related Questions

1. What happens to the maximum and minimum values if we change the function to a quadratic form?
2. How would the results differ if we considered a closed interval with endpoints that include critical points within the interval?
3. Can we generalize a method to find maximum and minimum values for any linear function over a given interval?
4. How does the slope of a linear function affect where the maximum and minimum occur?
5. What if the function was \( f(x) = 3x^2 + 3x \)? How would we find maximum and minimum values on intervals?

### Tip
For linear functions like \( f(x) = ax + b \), the maximum and minimum values on a closed interval always occur at the endpoints since the function has a constant rate of change with no turning points.

Find the absolute maximum and minimum values of the function f(x) = 3x + 3 over the intervals (A) [0,3] and (B) [-4,5], and indicate the x-values at which they occur.

Learn how to find the absolute maximum and minimum values of the function f(x) = 3x + 3 over the intervals [0,3] and [-4,5]. This solution covers linear functions, calculus concepts, and applies the Extreme Value Theorem. Suitable for high school calculus students in Grades 11-12.

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