To find relative extrema, we need to find P'(x) and solve for P'(x)=0.
P'(x)=150x-0.8x^3 [by the power rule]
Setting P'(x)=0 and solve for extrema. 150x-0.8x^3=0 => x(150-0.8x^2)=0 => 0.8x(187.5-x^2)=0 0.8x(5sqrt(15/2)-x)(5sqrt(15/2)+x)=0 => x={0,+5sqrt(15/2), -5sqrt(15/2)} by the zero product rule. [note: eqation P'(x)=0 can also be solved by the quadratic formula]
Reject negative root because we cannot hire negative persons.
So possible extrema are x={0,5sqrt(15/2)}
To find out which are relative maxima, we use the second derivative test. Calculate P"(x), again by the power rule, P"(x)=-1.6x For a relative maximum, P"(x)<0, so P"(0)=0 which is not <0 [in fact, it is an inflection point] P"(5sqrt(15/2))=-8sqrt(15/2) < 0, therefore x=5sqrt(15/2) is a relative maximum.
However, 5sqrt(15/2)=13.693 persons, which is impossible, so we hire either 13 or 14, but which one?
Let's go back to P(x) and find that P(13)=6962.8 P(14)=7016.8
So we say that assigning 14 employees will give a maximum output.
