The marketing research department of a computer company used a large city to test market the firm's new laptop. The department found the relationship between price \( p \) (dollars per unit) and the demand \( x \) (units per week) was given approximately by the following equation. \[ p=1620-0.15 x^{2} \quad 0

Ask by Ryan Mejia. in the United States

Mar 12,2025

To determine the concavity of the revenue function \( R(x) = 1620x - 0.15x^3 \), we need to calculate the second derivative of \( R(x) \). The first derivative is: \[ R'(x) = 1620 - 0.45x^2 \] The second derivative is: \[ R''(x) = -0.90x \] Now, let's analyze the second derivative, \( R''(x) = -0.90x \). The function is concave upward when \( R''(x) > 0 \) and concave downward when \( R''(x) < 0 \): A. The graph is concave upward on \( (0, 0) \) or, rather, there are no intervals where it is concave upward since \( R''(x) \) is never greater than 0 in the given domain of \( 0 < x < 80 \). B. The graph is never concave upward. As for concavity downward, since \( R''(x) < 0 \) for all \( x > 0 \) in the domain: A. The graph is concave downward on \( (0, 80) \). B. The graph is never concave downward is incorrect.