The total annual cost $C($$$)$ of an operating pump in a natural gas power plant is a funclion of the poner af the pump $x($horsepower$)$: $(10$ points$)$

$C=$$$500+$$$0\mathrm{.}9x+\frac{\text{\$}4500}{x}$

a$)$ Find the optimum $x$ that minimizes the total annual cost.

b$)$ Prove that this is the minimum point by taking the second derivative.

c$)$ Find the minimum $C.$