The function N(x) = -2x^3 + 180x^2 + 3000x + 20000 gives the number of items of a product that are sold for a given value of x,

Question

The function N(x) = −2x^3 + 180x^2 + 3000x + 20000
gives the number of items of a product that are sold for
a given value of x, where x represents advertising expenditures
in thousands of dollars. Find the value of x that represents
the point of diminishing returns.

Answer 1

To find the point of diminishing returns, we must find the maximum.  This is because if you are experiencing a diminish in returns, you are making more sales.  We will take the derivative of N(x) and set it equal to zero, since the slope of the line tangent to the maximum is zero.  Then solve for x.

 

N'(x) = 0

 

-6x² + 360x + 3000 = 0

 

-6(x² - 60x - 500) = 0

 

 

Set the factor in parenthesis equal to zero.

 

x² - 60x - 500 = 0

 

Use the quadratic formula.

 

x = (60 ± √(3600 - 4(-500))) / 2

x = (60 ± √(5600)) / 2

x = (60 ± 74.83) / 2

 

x =  (60 + 74.83) / 2            and         x = (60 - 74.83) / 2

 

x = 67.42                            and          x = -7.42

 

We have two x values.  This means that one of these will be x value of the maximum.  We can assume the positive value of x, but lets do a derivative test to make sure.

 

We will test these points into the derivative.  x =-8 , x = 1 , and x = 69.  If the derivative is negative, then the slope od the tangent line is negative.  If derivative is positive, slope is positive. 

 

N'(-8) = -6((-8)² - 60(-8) - 500)

         = -6(64 + 480 - 500)

         = -6(44)

 

N'(1) = -6((1)² - 60(1) - 500)

        = -6(1 - 60 - 500)

        = -6(-599)

 

N'(69) = -6((69)² - 60(69) - 500)

          = -6(4761 - 4140 - 500)

          = -6(121)

 

If the graph increases then decrease at a certain point, then we have a maximum.  This happens when x = 67.42 just as we have assumed before.

 

The x value that represents the point of diminishing returns is x = 67.42