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To solve the problem of finding the average rate of change of the function \( f(x) = 3x^2 - x^3 \) on the closed interval \([1, 5]\), we'll use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \(a = 1\) and \(b = 5\).

### Step 1: Find \( f(1) \)
\[
f(1) = 3(1)^2 - (1)^3 = 3 - 1 = 2
\]

### Step 2: Find \( f(5) \)
\[
f(5) = 3(5)^2 - (5)^3 = 3(25) - 125 = 75 - 125 = -50
\]

### Step 3: Calculate the Average Rate of Change
Now, we substitute these values into the formula:

\[
\text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{-50 - 2}{5 - 1} = \frac{-52}{4} = -13
\]

Thus, the average rate of change of \( f(x) \) on the interval \([1, 5]\) is \( \boxed{-13} \).

Would you like any further details or clarifications on this problem? 

Here are some follow-up questions that relate to the concept of average rate of change:
1. How does the average rate of change relate to the slope of the secant line between two points on the graph?
2. What is the difference between the average rate of change and the instantaneous rate of change?
3. Can you find the average rate of change on the interval \([0, 2]\) for the same function?
4. How would you interpret a positive versus a negative average rate of change in a real-world scenario?
5. How is the average rate of change related to the derivative of a function?

**Tip**: The average rate of change of a function over an interval gives the slope of the secant line that connects the two endpoints of the function on that interval.

If f is the function given by f(x) = 3x^2 - x^3, then the average rate of change of f on the closed interval [1, 5] is

Learn how to calculate the average rate of change for the polynomial function f(x) = 3x^2 - x^3 over the interval [1, 5]. This example includes step-by-step calculations and explanations, covering key concepts in calculus. Suitable for students in Grades 11-12.

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