suppose that the total profit in hundreds of dollars from selling x items is given by p(x)=4x^2 - 7x + 9…

suppose that the total profit in hundreds of dollars from selling x items is given by p(x)=4x^2 - 7x + 9. complete parts a through d below.\na. find the average rate of change of profit as x changes from 3 to 5.\n$ 2500 per item\nb. find the average rate of change of profit as x changes from 3 to 4.\n$ 2100 per item\nc. find and interpret the instantaneous rate of change of profit with respect to the number of items produced when x = 3. (this number is called the marginal profit at x = 3.)\n$ 1700 per item\nwhat does this result mean? choose the correct answer below.\na. when 3 items are sold, the profit is decreasing at the rate of $ per item.\nb. when items are sold for $, the profit is increasing at the rate of $3 per item.\nc. when items are sold for $, the profit is decreasing at the rate of $3 per item.\nd. when 3 items are sold, the profit is increasing at the rate of $ 1700 per item.\nd. find the marginal profit at x = 5.

suppose that the total profit in hundreds of dollars from selling x items is given by p(x)=4x^2 - 7x + 9. complete parts a through d below.\na. find the average rate of change of profit as x changes from 3 to 5.\n$ 2500 per item\nb. find the average rate of change of profit as x changes from 3 to 4.\n$ 2100 per item\nc. find and interpret the instantaneous rate of change of profit with respect to the number of items produced when x = 3. (this number is called the marginal profit at x = 3.)\n$ 1700 per item\nwhat does this result mean? choose the correct answer below.\na. when 3 items are sold, the profit is decreasing at the rate of $ per item.\nb. when items are sold for $, the profit is increasing at the rate of $3 per item.\nc. when items are sold for $, the profit is decreasing at the rate of $3 per item.\nd. when 3 items are sold, the profit is increasing at the rate of $ 1700 per item.\nd. find the marginal profit at x = 5.

Answer

Explanation:

Step1: Find the derivative of the profit - function

The profit function is $P(x)=4x^{2}-7x + 9$. Using the power - rule for differentiation $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, the derivative $P^\prime(x)=\frac{d}{dx}(4x^{2}-7x + 9)=8x-7$.

Step2: Calculate the marginal profit at $x = 5$

Substitute $x = 5$ into $P^\prime(x)$. We have $P^\prime(5)=8\times5-7$. $P^\prime(5)=40 - 7=33$. Since the profit function $P(x)$ is in hundreds of dollars, the marginal profit at $x = 5$ is $33\times100=$3300$ per item.

Answer:

$3300$ per item