We are given the profit function $P(x) = 3x^2 - 4x + 9$, where $P(x)$ represents the total profit in hundreds of dollars from selling $x$ items. To find the average rate of change of profit, we will use the formula for the average rate of change over an interval $[a, b]$:

$\text{Average rate of change} = \frac{P(b) - P(a)}{b - a}$

Part 1: $x$ changes from 2 to 4

We will calculate the average rate of change of profit as $x$ changes from 2 to 4.

1. Compute $P(2)$ and $P(4)$: $P(2) = 3(2)^2 - 4(2) + 9 = 3(4) - 8 + 9 = 12 - 8 + 9 = 13$ $P(4) = 3(4)^2 - 4(4) + 9 = 3(16) - 16 + 9 = 48 - 16 + 9 = 41$

2. Calculate the average rate of change: $\frac{P(4) - P(2)}{4 - 2} = \frac{41 - 13}{4 - 2} = \frac{28}{2} = 14$

So, the average rate of change of profit as $x$ changes from 2 to 4 is 14 hundred dollars per item, or $1400 per item.

Part 2: $x$ changes from 2 to 3

Next, we will calculate the average rate of change of profit as $x$ changes from 2 to 3.

1. Compute $P(3)$: $P(3) = 3(3)^2 - 4(3) + 9 = 3(9) - 12 + 9 = 27 - 12 + 9 = 24$

2. Calculate the average rate of change: $\frac{P(3) - P(2)}{3 - 2} = \frac{24 - 13}{3 - 2} = \frac{11}{1} = 11$

So, the average rate of change of profit as $x$ changes from 2 to 3 is 11 hundred dollars per item, or $1100 per item.

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Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How does the average rate of change relate to the derivative of the profit function?
2. What is the instantaneous rate of change of profit at $x = 3$?
3. How would the average rate of change change if $x$ were increased to 5?
4. How can you interpret the values of $P(x)$ economically?
5. What is the significance of the coefficients in the quadratic profit function?

Tip: The average rate of change of a function over an interval provides a good estimate of the function's behavior between two points, similar to the slope of a line connecting them.