The weekly revenue from the production and sale of units of grain is given by thousand dollars. The cost function is given by
thousand dollars. Find the number of units of grain that are to be produced to maximize the profit if .

To find the number of units of grain that maximize the profit, we start by defining the profit function as . Substituting the given revenue and cost functions, we get:

Simplifying this, we combine like terms:

To find the maximum profit, we take the derivative of :

Setting this derivative to zero to find critical points:

We also check the endpoints of the interval to determine the maximum profit:

Comparing the profit amounts, we find that is indeed the maximum. Therefore, the number of units of grain that should be produced to maximize profit is .

Understanding profit maximization goes beyond just mathematical calculations, it has historical roots in economic theory! In the early 19th century, economists like Adam Smith and David Ricardo began to formalize concepts of profit and competitive markets, paving the way for modern economic practices and theories that help businesses make informed decisions today.

In real-world scenarios, maximizing profit isn’t just about crunching numbers; it also involves strategic decision-making! Companies often utilize profit maximization models along with market research, production capabilities, and consumer demand analyses. This blend of math and practical insight ultimately guides firms to make better production choices and improve overall profitability in competitive markets.