1. **State the problem:** We have the profit function $$P(x) = 5x^2 - 5x + 9$$ and need to find the average rate of change of profit between certain values of $x$, and the instantaneous rate of change (marginal profit) at $x=2$. 2. **Formula for average rate of change:** The average rate of change of a function $P(x)$ from $x=a$ to $x=b$ is given by: $$\text{Average rate of change} = \frac{P(b) - P(a)}{b - a}$$ This measures the change in profit per item as production changes from $a$ to $b$. 3. **Calculate average rate of change from $x=2$ to $x=4$: ** Calculate $P(2)$ and $P(4)$: $$P(2) = 5(2)^2 - 5(2) + 9 = 5 \times 4 - 10 + 9 = 20 - 10 + 9 = 19$$ $$P(4) = 5(4)^2 - 5(4) + 9 = 5 \times 16 - 20 + 9 = 80 - 20 + 9 = 69$$ Now compute average rate: $$\frac{P(4) - P(2)}{4 - 2} = \frac{69 - 19}{2} = \frac{50}{2} = 25$$ 4. **Calculate average rate of change from $x=2$ to $x=3$: ** Calculate $P(3)$: $$P(3) = 5(3)^2 - 5(3) + 9 = 5 \times 9 - 15 + 9 = 45 - 15 + 9 = 39$$ Average rate: $$\frac{P(3) - P(2)}{3 - 2} = \frac{39 - 19}{1} = 20$$ 5. **Find instantaneous rate of change (marginal profit) at $x=2$: ** The instantaneous rate of change is the derivative $P'(x)$ evaluated at $x=2$. Find derivative: $$P'(x) = \frac{d}{dx}(5x^2 - 5x + 9) = 10x - 5$$ Evaluate at $x=2$: $$P'(2) = 10(2) - 5 = 20 - 5 = 15$$ 6. **Interpretation:** The marginal profit at $x=2$ is 15, meaning the profit is increasing by 15 units per additional item produced at $x=2$. 7. **Match to answer choices:** Since profit is increasing at $x=2$ by 15 units per item, the correct choice is: B. When 2 items are sold, the profit is increasing at the rate of 15 per item. **Final answers:** - Average rate of change from 2 to 4: 25 - Average rate of change from 2 to 3: 20 - Instantaneous rate of change at 2: 15 - Interpretation: Choice B Note: The problem states values like 2500, 2000, 1500 per item, but based on the function given, the calculated rates are 25, 20, and 15 respectively. Possibly the units are scaled by 100, so multiply by 100 to match given values if needed.