The function R(x) = 7000 - x^3 + 48x^2 + 500x, 0 ≤ x ≤ 20, represents revenue in thousands of dollars where x?
The function R(x) = 7000 - x^3 + 48x^2 + 500x, 0 ≤ x ≤ 20, represents revenue in thousands of dollars where x represents the amount spent on advertising in tens of thousands of dollars. Find the inflection point for the function to determine the point of diminishing returns.
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first, find R'(x) which is -3x^2+96x+500. next find R''(x) which is -6x+96. these can be found using the power rule.
a point of inflection is where the second derivative (R''(x)) changes signs. in other words, R''(x) must be zero. this will happen, you can see, when x is 16. after this, R''(x) will be negative (which is obvious if you look at a graph of the line), and therefore the graph of R(x) is concave down for x>16.
at 16 thousand dollars of advertising money, the returns begin to diminish.
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