Question

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.
a. find the average rate of change of profit as x changes from 3 to 5.
$ 2500 per item
b. find the average rate of change of profit as x changes from 3 to 4.
$ 2100 per item
c. 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.)
$ 1700 per item
what does this result mean? choose the correct answer below.
a. when 3 items are sold, the profit is decreasing at the rate of $ per item.
b. when items are sold for $, the profit is increasing at the rate of $3 per item.
c. when items are sold for $, the profit is decreasing at the rate of $3 per item.
d. when 3 items are sold, the profit is increasing at the rate of $ 1700 per item.
d. find the marginal profit at x = 5.

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