Find the maximum profit and the number of units that must be

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit Assume that revenue. R(x), and cost, C(x), are in thousands of dollars, and x is in thousands of units. R(x() = 8x - 4x^2, C(x) = x^3 - 5x^2 + 4x + 1 The production level for the maximum profit is about units. (Do not round unit the final answer Then round to the whole number as needed)

Solution

R(X) = -4x^2 +8x ; C(x) = x^3 - 5x^2 +4x +1

Profit P(x) = R(x) - C(x) = -4x^2 +8x - x^3 +5x^2 -4x -1

= -x^3 +x^2 +4x -1

Find maximum of P(x) : dP/dx = - 3x^2 + 2x + 4

Solve for - 3x^2 + 2x + 4 =0

x = ( -2 +/- sqrt(4 +48) )/-6 = ( 1 +/- sqrt13)/3

find f\"(x) = -6x +2 ; f\"( (1 -sqrt13)/3) = -7.21 minoium occurs

So, maximum occurs at x = ( sqrt13 +1)/3 = 1.535 = 2

Solution   : 2000 units