Final answer:
To determine the unit price that yields maximum profit for the given commodity with specified demand and cost functions, calculate the profit function, find its derivative with respect to the number of units, set it to zero, and solve for the quantity that maximizes profit. Then, use this quantity to find the profit-maximizing price.
Step-by-step explanation:
The goal is to find the unit price that yields maximum profit for a commodity with a demand function p = 112 - 0.5x and a total cost function C = 50x + 31.75, where x is the number of units.
To determine the profit-maximizing price, we first need to find the profit function, which is the total revenue minus the total cost: Profit = Revenue - Cost. Total revenue (Revenue) is calculated as the product of the demand function, p, and the number of units sold, x, which gives Revenue = p*x = (112 - 0.5x)*x. The profit function is therefore Profit = (112 - 0.5x)*x - (50x + 31.75).
To find the maximum profit, we need to calculate the derivative of the profit function with respect to x and set it to zero, then solve for x. The value of x that maximizes profit can then be substituted back into the demand function to find the profit-maximizing price.