Answer
$p(x)=\frac{2}{3}x\sqrt{x}+\frac{1}{2}x-1000$
Work Step by Step
The profit function is:
$$p(x)=\int p'(x)dx$$
$$p(x)=\int \left(\sqrt{x}+\frac{1}{2}\right)dx$$
$$p(x)=\frac{2}{3}x\sqrt{x}+\frac{1}{2}x+C$$
It is given that for $x=0$ we have $p(0)=-1000$.
So:
$$p(0)=\frac{2}{3}\sqrt{0^{3}}+\frac{1}{2}\cdot 0+C$$
$$-1000=\frac{2}{3}\sqrt{0^{3}}+\frac{1}{2}\cdot 0+C$$
$$-1000=C$$
Therefore, the profit function is:
$$p(x)=\frac{2}{3}x\sqrt{x}+\frac{1}{2}x-1000$$
