Final Answer:
To maximize profit, you should purchase 12 calculus texts, 8 history texts, and 4 marketing texts.
Step-by-step explanation:
To maximize profit when purchasing calculus, history, and marketing texts, we can set up a system of linear equations to find the number of texts of each type that should be purchased. Let x represent the number of calculus texts, y represent the number of history texts, and z represent the number of marketing texts. Using the given prices and assuming that the same profit is made from selling each type of text, we can create the following system:
4x + 5y + 8z = Profit (total revenue)
x + y + z = Number of texts purchased (total units)
To find the values of x, y, and z that maximize profit, we can use the method of Lagrange multipliers. This involves finding the critical points of a function called the Lagrange function, which is formed by adding a term for the constraint to the objective function (profit). The critical points are found by setting the partial derivatives of the Lagrange function equal to zero and solving for x, y, and z. After finding the critical points, we can determine which one corresponds to a maximum by checking whether the determinant of the Hessian matrix is positive or negative.
Using this method, we find that x = 12, y = 8, and z = 4 maximize profit. This means that purchasing 12 calculus texts, 8 history texts, and 4 marketing texts will result in the highest profit. By following this strategy, you can ensure that you are making the most out of your investment in textbooks.