Question

The total profit made by an engineering firm is given by the function p=x^2 - 24x +5000 where x is the number of clients the firm has and p is the profit.
1. Find the maximum profit made by the company.

2. How many clients are necessary to reach the profit outlined in part one?

Answer 1

Answer:

$5144$

$12$

Step-by-step explanation:

The function $p=x^2-24x+5000$ is incorrect as its roots are imaginary $b^2-4ac=576-4\times 5000=-19424$.

So, the correct function is

$p=-x^2+24x+5000$

Differentiating with respect to $x$ we get

$p'=-2x+24$

Equating with zero

$0=-2x+24\\\Rightarrow x=\dfrac{24}{2}\\\Rightarrow x=12$

Double derivative of the function

$p''=-2$

So, the function is maximum at $x=12$

Maximum profit is

$p=-x^2+24x+5000=-12^2+24\times 12+5000\\\Rightarrow p=5144$

The maximum profit made by the company is $5144$

The number of clients required to make the maximum profit is $12$.