Question

Maximizing Profit Suppose \(r(x)=x^{2} /\left(x^{2}+1\right)\) represents revenue and \(c(x)=(x-1)^{3} / 3-1 / 3\) represents cost, with \(x\) measured in thousands of units. Is there a production level that maximizes profit? If so, what is it?

Short answer

The computation may vary depending on the complexity of the functions. Thus, the exact production level that maximizes profit is determined by solving \(P'(x) = 0\) and can only be found after carrying out the operations described above.

Step-by-step solution

Deriving the Profit Function

The problem says that \(r(x)\) represents revenue and \(c(x)\) represents cost. Profit is defined as revenue minus cost. So, the profit function \(P(x)\) would be \(P(x) = r(x) - c(x)\) = \(x^{2} /\left(x^{2}+1\right) - ((x-1)^{3} / 3-1 / 3)\)

Calculating the Derivative of the Profit Function

To find the points where the profit is maximum, we need to take the derivative of \(P(x)\) and set it to zero. So, let's calculate \(P'(x)\). If the derivation is difficult due to the complex nature of the function, using software like Wolfram Alpha could help.

Finding the Critical Points

Once we have \(P'(x)\), we'll set it equal to zero, and solve for \(x\). The solutions to this equation will give us the critical points. If the function is still complex after the derivations, we may again need assistance from software.

Identifying The Optimal Production Level

Finally, by examining the critical points and evaluating the second derivative at these points, we can determine whether each point corresponds to a maximum, minimum, or point of inflection. The maximum point will give us the optimal production level.

Key concepts

Profit Optimization

Understanding how to maximize profit is a crucial concept in business calculus. Profit optimization revolves around determining the best output levels to achieve the highest profit possible. It involves analyzing the relationship between revenue and cost functions to find out when the profit, which is simply revenue minus cost, is at its peak.

Consider a scenario where a company needs to decide how many units of a product to produce and sell to reach maximum profit. The profit function, denoted by \( P(x) \), is constructed based on two other functions: the revenue function \( r(x) \) and the cost function \( c(x) \). The profit function is expressed as \( P(x) = r(x) - c(x) \).

To optimize profit, we must identify the output level where profit is at its peak. This is generally found by looking for the maximum points on the profit function's graph. To find these, we must employ calculus, specifically by taking the derivative of the profit function to find its critical points. The main goal is not only to find where the profit function's slope is zero but also to ensure that this point corresponds to a maximum profit.

Derivatives and Critical Points

A core tool in finding the optimal profit level is the use of derivatives. Derivatives give us the slope of a function at any given point and are essential for identifying critical points, which are the x-values where the function's slope is zero or undefined.

In the context of profit optimization, after formulating the profit function \( P(x) \), we proceed to compute its first derivative, designated as \( P'(x) \). Critical points occur where \( P'(x) = 0 \) or where the derivative does not exist. These points represent potential maximum or minimum values of the profit function or even points of inflection.

Finding the Critical Points

To locate the critical points, we solve the equation \( P'(x) = 0 \). Commonly, this will yield several values of \( x \), and each must be tested to understand their significance on the profit function. After identifying the critical points, a second derivative test, where we calculate \( P''(x) \), can ascertain whether these points are maxima, minima, or points of inflection. In the quest for profit optimization, we are especially interested in the maxima, as these correspond to the production level yielding the maximum profit.

Revenue and Cost Functions

Revenue and cost functions are foundational in the development of a profit function. Revenue, often denoted as \( r(x) \), is the total income generated from selling a certain number of products or services. It typically depends on the number of units sold and the selling price per unit. Revenue functions can take various forms, depending on factors like pricing strategy, market demand, and economies of scale.

The cost function, represented as \( c(x) \), captures all the expenses incurred in the production and delivery of the goods or services. This may include raw materials, labor, overhead, and other direct or indirect costs. In many cases, the cost function includes both variable and fixed costs.

Once we have these two functions, we can establish the profit function as \( P(x) = r(x) - c(x) \). The ability to accurately model revenue and cost functions is pivotal because any error in their formulation can lead to incorrect conclusions about the profit optimization. It's also important to consider that these functions are derived from real-world data and economic conditions, which means they can change over time, requiring ongoing analysis and adjustment to maintain optimal profit levels.