Business Applications of Extrema

The video begins with an introduction to the business applications of finding extrema, which are the maximum and minimum values of a function. The presenter explains the use of the first derivative test to identify these values in a business context, such as maximizing profit or minimizing cost. Key business functions like demand, cost, revenue, and profit are defined in relation to the number of units (X). The concept of average cost, revenue, and profit is also introduced, explaining how they are calculated by dividing the respective total function by X. An example of a cubic revenue function is provided to illustrate the process of finding the production level that yields maximum revenue.

The presenter discusses how to find the production level that maximizes revenue using the first derivative test. By differentiating the revenue function with respect to the number of items produced and sold (X), critical numbers are identified. The video demonstrates solving a quadratic equation to find the critical points and using the first derivative test to confirm that the critical point corresponds to a maximum revenue. The example provided involves a company with a cubic revenue function, and the process of finding the maximum revenue and the corresponding production level is shown step by step.

The focus shifts to finding the production level that minimizes the average cost of producing units. The presenter outlines the process of deriving the average cost function by dividing the total cost function by X and then differentiating it to find the minimum. The example given involves a quadratic cost function, and the presenter shows how to rewrite the average cost function, take its derivative, and solve for critical numbers to find the production level that minimizes the average cost. The domain of the problem is considered to ensure that the solution is applicable to the business context.

The video tackles a more complex problem where the presenter must create equations based on given information to maximize monthly revenue. The business scenario involves a fixed number of units sold per month at a certain price, with the potential to sell more units with a price reduction. Using two points (price and quantity combinations), a price function is derived, which is then used to express revenue as a function of X. The presenter differentiates the revenue function to find the critical points and uses these to identify the price that maximizes monthly revenue. The process involves careful consideration of the relationship between price, quantity, and revenue.

The final example is about maximizing profit, which requires creating a profit function from the given demand and cost functions. The presenter explains that profit is the difference between revenue (price times quantity) and cost. After expressing the profit function in terms of X, the derivative is taken to find the critical points that represent the maximum profit. The presenter shows how to solve for the critical points by setting the derivative equal to zero and using algebraic manipulation. The example involves a demand function that relates price to the square root of quantity and a linear cost function, leading to the determination of the price that yields maximum profit.

The presenter concludes by emphasizing the importance of understanding what is being maximized or minimized and focusing on the correct function to derive. They highlight that sometimes the answer involves plugging the critical value back into the original function to find the actual maximum or minimum value. The presenter offers help for further questions and encourages viewers to apply the concepts learned to their assignments, ensuring they read the questions carefully to provide the correct answer.