Question

You are the marketing executive for a tech firm that sells two different competing products. The company produces x inousand units of product #1 and y thousand units of product $2. You have done some data analytics and have determined that over the course of a year, the revenue equation, in milisons of dollars, is: R(x,y)=4x+3y You have also determined that the costs incurred for manufacturing these products per year is C(x,y)=x²−3xy+7y² +2x−13y−4 (in millions or dollars). Determine how many units of each product should be sold per year to maximize the tech firm s profit

Answer 1

To maximize the tech firm's profit, the number of units to be sold per year for product #1 (x) and product #2 (y) should be determined by optimizing the profit function. The profit function P(x, y) can be expressed as the difference between the revenue function R(x, y) and the cost function C(x, y): P(x, y) = R(x, y) - C(x, y) = (4x + 3y) - (x² - 3xy + 7y² + 2x - 13y - 4).

Step-by-step explanation:

To find the maximum profit, we need to identify the critical points of the profit function P(x, y) by taking partial derivatives with respect to x and y and setting them equal to zero. Solving the resulting system of equations will give the values of x and y at which the profit is maximized. Once the critical points are found, additional analysis using the second partial derivative test ensures that these points correspond to a maximum rather than a minimum.

After obtaining the critical points and verifying their nature, the next step is to evaluate the profit function at these points. The resulting values will indicate the optimal number of units for product #1 (x) and product #2 (y) that maximize the tech firm's profit. This mathematical approach enables the marketing executive to make informed decisions regarding the product quantities to achieve the highest profitability for the tech firm.