(a) The monthly revenue is given by the function [tex]R(x) = x \cdot p(x)[/tex], and we are given that [tex]p(x) = 5 - 0.001x[/tex]. So, we have
[tex]R(x) = x (5 - 0.001x) = 5x - 0.001 x^2[/tex].
(b) The monthly profit is given by the function [tex]P(x) = R(x) - C(x)[/tex], and we are given that [tex]C(x) = 35 + 1.5x[/tex]. So, we have
[tex]P(x) = (5x - 0.001x^2) - (35 + 1.5x) = 3.5x - 0.001x^2 - 35[/tex].
(c) We already have the formula for P(x), so you can just plug in the values:
x = 600, x = 1200, x = 1800, etc. into P(x) and write down the answers.
The trickier part is finding the derivatives. We'll start with dR/dx:
[tex]\frac{dR}{dx} = \frac{d}{dx}[5x - 0.001x^2] = 5 - 0.002x[/tex].
The next is dP/dx:
[tex]\frac{dP}{dx} = \frac{d}{dx}[3.5x - 0.001x^2 - 35] = 3.5 - 0.002x - 0 = 3.5 - 0.002x[/tex].
Once again, just plug in the different values of x into the formulas we found and we will be able to complete the table.
If you're confused on how I calculated the derivatives, let me know. I made the assumption that you already knew how to do that (although that is not always true). :)
