Calculating Karim's Optimal Consumption with a Wage of €30

The Second Derivative Test for Maximization

Profit Maximization Analysis

A firm's profit (Π) is described by the function Π(Q) = -2Q² + 80Q - 150, where Q is the quantity of output. Using calculus, determine the quantity (Q) that maximizes the firm's profit.

A company's cost function is given by C(q) = q³ - 6q² + 50, where q is the quantity produced (q > 0). A manager identifies that q = 4 is a stationary point because setting the first derivative of the cost function to zero yields this value. The manager concludes that producing 4 units will therefore maximize the company's cost. Is this conclusion correct?

Verifying an Optimal Point

You are given a function that describes a firm's profit in terms of a single variable, such as the quantity of output. To find the specific value of the variable that results in the highest possible profit, you must follow a specific analytical procedure. Arrange the following steps in the correct logical order to identify and confirm a profit-maximizing point.

Sufficiency of Optimization Conditions

For each of the following single-variable functions, find the stationary point by setting the first derivative equal to zero. Then, use the second derivative to determine the nature of this point. Match each function to the correct description of its stationary point.

A company's total revenue (R) from selling a product is given by the function R(q) = 400q - 2q², where q is the number of units sold. To achieve the highest possible revenue, the company should sell ____ units.

Evaluating an Analyst's Profit Maximization Claim

An analyst is examining a firm's profit function, Π(Q), where Q is the output level. At a specific output level Q*, the analyst finds that the first derivative is zero (Π'(Q*) = 0) and the second derivative is positive (Π''(Q*) > 0). What is the correct interpretation of these findings regarding the output level Q*?