Given the following revenue and cost functions, find the x-value that makes profit a maximum. R(x) = 58x - 2x2; C(x) = 21x + 109

Question

solve the problem

Answer 1

R(x) = 58x - 2x²

C(x) = 21x + 109

 

The Profit, P(x), is Revenue - Costs:

 

P(x) = -2x² + 58x - 21x - 109

P(x) = -2x² + 37x - 109

 

P(x) is a quadratic equation whose graph is a parabola.  Since the coefficient of the x² term (-2) is less than zero, it's an inverted parabola with the vertex at the top.  So the vertex is maximum point on the profit curve, and represents the maximum profit.  The vertex is always located at the point x = -b/2a where b is the coefficient of the x term (37) and a is the coefficient of the x² term (-2).  So the value of x that yield the maximum profit is:

 

x = -b/2a = -37/2(-2) = 37/4 = 9 ¹/₄