The profit is a max (or min) if `(dP)/(dx)=0`.
Is it a maximum?
`(d^2 P)/(dx^2) = -0.04 < 0` for all x, so we have a maximum.
When `x = 200`, `P = $800`.
So if the company refines `200` barrels per day, the maximum profit of `$800` is reached.
Graph of `P=8x-0.02x^2`.
The maximulm point, `(200, 800)` is indicated on the graph with a magenta dot.