Final answer:
The point of diminishing returns for the revenue function R(x) = -8x^3 + 72x^2 + 60x + 40 is determined by the second derivative test. However, the second derivative of the function does not change from positive to negative in a realistic, positive range of x (which is in thousands of dollars), so the function doesn't have a point of diminishing returns for x > 0.
Step-by-step explanation:
The question you've asked pertains to finding the point of diminishing returns for advertising costs, which is a calculus concept that looks for a point at which the rate of increase of revenue (R(x)) begins to decline, that is, when the second derivative of R(x) becomes negative. The revenue function provided is R(x) = −8x³ + 72x² + 60x + 40, where x represents thousands of dollars. To find the point of diminishing returns, we need to calculate the first and second derivatives of R(x), and look for the value of x where the second derivative changes sign (from positive to negative).
First, we find the first derivative of R(x): R'(x) = ∘24x² + 144x + 60. Then we find the second derivative: R''(x) = ∘48x +144. The point of diminishing returns occurs where R''(x) = 0. Solving ∘48x +144 = 0 gives x = -3; however, since negative amounts of spending do not make sense in this context, we would look for the nearest interval in which sign changes from positive to negative for the second derivative in the relevant domain.
Since R''(x) = ∘48x +144 is a linear function with a negative slope, as x increases, R''(x) becomes less. The transition from positive to negative value shows the inflection point, which is the point of diminishing returns. With the calculated x value not being applicable, we understand that for positive values of x, the second derivative remains positive. Therefore, in this case, the point of diminishing returns does not exist within a realistic, positive range of x representing thousands of dollars in advertising costs.