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Let's solve part (c) for the given problem.

The function is \( f(x) = 3x^2 + 6 \), and we are asked to find the **average rate of change** of the function over the interval [2, 5].

The formula for the average rate of change between two points \( x_1 \) and \( x_2 \) is:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

For part (c), the interval is from 2 to 5, so:

1. First, calculate \( f(2) \):
   \[
   f(2) = 3(2)^2 + 6 = 3(4) + 6 = 12 + 6 = 18
   \]

2. Now calculate \( f(5) \):
   \[
   f(5) = 3(5)^2 + 6 = 3(25) + 6 = 75 + 6 = 81
   \]

3. Finally, apply the average rate of change formula:
   \[
   \frac{f(5) - f(2)}{5 - 2} = \frac{81 - 18}{5 - 2} = \frac{63}{3} = 21
   \]

Thus, the average rate of change from 2 to 5 is **21**.

Would you like more details or have any questions?

Here are 5 related questions to think about:
1. How is the average rate of change related to the derivative?
2. What is the geometric interpretation of the average rate of change?
3. How would the average rate of change differ for a linear function?
4. Can the average rate of change be negative? Why?
5. How would you find the instantaneous rate of change at a specific point?

**Tip:** The average rate of change can be visualized as the slope of the secant line connecting two points on the graph of the function.

Find the average rate of change of f(x) = 3x^2 + 6 over the intervals (a) from -2 to 0, (b) from -1 to 1, and (c) from 2 to 5.

Learn how to find the average rate of change of the quadratic function f(x) = 3x^2 + 6 over different intervals. This problem covers key algebraic concepts and uses the formula for the average rate of change. Suitable for high school students in Grades 9-12.

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