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 $x = -4$

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?

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.