The total profit Px in thousands of dollars from a sale of x

The total profit P(x) (in thousands of dollars) from a sale of x thousand units of a new product is given by P(x) = In ( - x3 + 12x2 + 27x + 1) . a) Find the number of units that should be sold in order to maximize the total profit, b) What is the maximum profit?

Solution

for max profit P\'(x)=0 ==>(-3x^2+24x+27)/(-x^3+12x^2+27x+1)=0 ==>x^2-8x-9=0 ==>x=9------------->number of units maximium profit=ln(-9^3+12*9^2+27*9+1) =ln 487 =6186.2$