Question

1. The profit of a company is given by P(x) = 5 000 +1 000x-5x² where x is the amount (in thousands of pesos) that the company spends on advertising.

a. Find the amount x that the company has to spend to maximize its profit.

b. Find the maximum profit.

Answer 1

$$P(x) = 5 000 +1 000x-5x^2, x\geq 0$$

Find the first derivetive with respect to $x$

$$P'(x)=1000-10x$$

Find the critical number(s)

$$P'(x)=0=>1000-10x=0$$

$$x=100$$

If $0<x<100, P'(x)>0, P(x)$ increases.

If $x>100, P'(x)<0, P(x)$ decreases.

The function $P(x)$ has a local maximum at $x=100.$

Since the function $P$ has the only extremum, then the function $P(x)$ has the absolute maximum at $x=100.$

a. The company has to spend $100,000$ pesos has to maximize its profit.

b.

$$P(100) = 5 000 [phone](100)^2$$

$$=55000$$

The maximum profit is $55000.$