The total annual revenue R in thousands of hotel is related

The total annual revenue, R (in thousands) of hotel is related to the amount of money, x(in thousands of dollars) spent on advertising by the function R(x) = - 0.02x^3 + 1.5x^2 - x + 500 (0 lessthanorequalto x lessthanorequalto 300) Find the inflection point and explain what this tells the hotel owners about the revenue using the context of the problem. Write out your answer in sentences that give the context of the answer.

Solution

Solution:

An inflection point is a point on the graph where there is a change in concavity.

We thus need the second derivative R¢¢(x).

The first is R¢(x) = -0.06 x² + 3x - 1, and the second is

  R¢¢(x) = - 0.12 + 3

Setting -0.06x + 3 = 0 gives x = 150. The sign graph of the second derivative is

We see that R changes from concave up to concave down as we move across x=150.

Therefore,

the point (150, 28550) is the inflection point of the graph of R(x).

The revenue is increasing at an increasing rate until the amount of money spent on advertising is $100,000.

After that, the revenue is still increasing but at a decreasing rate.