I dont understand how to find the maximum revenue of the equation R(x)=2000x-2xsquared

Question

x= the number of items produced during a week

Answer 1

R = -2x² + 2000x

 

There are two ways to find the maximum revenue, using calculus and using algebra.

 

CALCULUS

 

Take the derivative of R wrt x, set it to zero, and solve for x.  Setting the derivative to zero will find the extreme points (maximums and/or minimums) of the function.

 

dR/dx = -4x + 2000

0 = -4x + 2000

4x = 2000

x = 500

 

Plug x = 500 into the revenue equation to get the max revenue.

 

ALGEBRA

 

The revenue equation is a quadratic, so its graph is a parabola.  Since the coefficient of the x² term is negative (-2), it's an inverted parabola with the vertex at the top.  The vertex will thus be the maximum revenue.  To find the vertex, convert R to the vertex form R = a(x-h)²+k where (h,k) is the location of the vertex.  Convert to the vertex form by completing the square:

 

R = -2x² + 2000x

R = -2(x² - 1000x + 250,000) + 500,000

R = -2(x-500)² + 500,000

 

The vertex is located at x=500 and R = $500,000.  The max Revenue is thus $500,000.