Graph the marginal profit function for the profit function P(x) = 11x − 0.4x2 − 10, where P(x) is in hundreds of dollars and x is hundreds of units.

Question

Graph the marginal profit function for the profit function P(x) = 11x − 0.4x² − 10,

 where P(x)  is in hundreds of dollars and x is hundreds of units.

(b) What level of production and sales (in units) will give a 0 marginal profit?

(c) At what level of production and sales (in units) will profit be at a maximum?

(d) What is the maximum profit (in dollars)?

Answer 1

Given the profit function, the marginal profit function is the derivative of the profit function with respect to an infinitesimal change in the profit function

M(x) = P'(x) = 11 - 0.4*2*x = 11 - 0.8x

Knowing the marginal profit, we can now set it to zero to find the number of units...

M(x) = 0 ---> 11 = 0.8x ---> x = 110/8

Maximal profit will be the critical point (zero of marginal profit function when P"(x) < 0. So...

P"(x) = M'(x) = -0.8

Since it is alwats negative then the value of x which makes M(x) = 0 is the value for maximal profit... x = 110/8

The largest profit is

P(x)|x= 110/8 ---> P(110/8) = 11*(110/8) - 0.4(110/8)^2 - 10

Max profit = $151.25 - $75.625 - $10 = $65.625