Answer:
50 (hundred dollars)
Explanation:
To get maximum profit, we differentiate P and the result, P', we set equal to zero.
[tex]P(x)=200+50x-0.5x^{2}\\ \\P'(x)=50-x\\\\0=50-x\\\\x=50[/tex]
Every time you need to optimize a function, you differentiate and equal to zero to get the answer or answers!
In this case if we plug 50 into P we get:
[tex]P(x)=200+50x-0.5x^{2}\\\\P(50)=200+50(50)-0.5(50)^{2}\\\\P(50)=200+2500-1250\\\\P(50)=1450\\\\[/tex]
Remember that the function yields results in hundred of dollars, so the profit would be equal to 145000 dollars.
Look at the graph of the P function, at x= 50 we see the vertex of the parabola which is the highest point over the entire graph, which is also its maximum profit or value.
