A company uses the model R(x) = (600x2-x3)/20000

Question

A company uses the model

R(x) = (600x²-x³)/20000

To relate its revenue R (in thousands of dollars) to the amount of money x (in thousands of dollars) it spends on advertising. Find the point of diminishing returns for this model.

Answer 1

To find the point of diminishing returns, we need to determine the value of x, where the rate of increase in revenue R(x) starts to slow down. It happens when the second derivative of the revenue function changes from positive to negative.

R'(x) = (1200x - 3x²) / 20000

R''(x) = (1200 - 6x) / 20000

Let's equal to zero R''(x):

1200 - 6x =0 -> x = 200

So, the point of diminishing returns for this model occurs when the company spends $200,000 on advertising. At this level of advertising expenditure, the revenue increase starts to slow down.