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Chapter 7: Problem 7

Reformulate the problem of finding a minimum or maximum for a function\(D\left(x_{1}, x_{2}\right)\) as a rootfinding problem for a system of twoequations in two unknowns. We assume the function \(D\left(x_{1}, x_{2}\right)\)is differentiable with respect to both \(x_{1}\) and \(x_{2}\).

### Short Answer

Expert verified

Set \( \frac{\partial D}{\partial x_1} = 0 \) and \( \frac{\partial D}{\partial x_2} = 0 \). Solve this system for \( x_1 \) and \( x_2 \).

## Step by step solution

01

## - Define the Objective

The objective is to reformulate the problem of finding the minimum or maximum of the function \( D(x_1, x_2) \).

02

## - Identify the Necessary Condition

To find the minimum or maximum of the function \( D(x_1, x_2) \), the necessary condition is that the partial derivatives with respect to both variables \( x_1 \) and \( x_2 \) must be zero.

03

## - Set Up the Partial Derivatives

Calculate the partial derivatives of \( D \) with respect to \( x_1 \) and \( x_2 \): \( \frac{\partial D}{\partial x_1} \) and \( \frac{\partial D}{\partial x_2} \).

04

## - Formulate the System of Equations

Set the partial derivatives equal to zero to form a system of two equations: \[ \frac{\partial D}{\partial x_1} = 0 \] and \[ \frac{\partial D}{\partial x_2} = 0 \].

05

## - Reformulate as a Rootfinding Problem

We now need to find the roots of the system of equations: \[ \frac{\partial D}{\partial x_1}(x_1, x_2) = 0 \] and \[ \frac{\partial D}{\partial x_2}(x_1, x_2) = 0 \]. This is the reformulated problem where we solve for \( x_1 \) and \( x_2 \) such that both equations are satisfied.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Partial Derivatives

To understand how to find the minimum or maximum of the function, we first need to explore partial derivatives. A partial derivative represents the rate of change of a function with respect to one of its variables while keeping other variables constant.

For a function like \( D(x_1, x_2) \), the partial derivative with respect to \( x_1 \) is denoted as \( \frac{\partial D}{\partial x_1} \). Similarly, the partial derivative with respect to \( x_2 \) is written as \( \frac{\partial D}{\partial x_2} \). These partial derivatives help us determine how \( D \) changes when we tweak either \( x_1 \) or \( x_2 \).

The key idea here is that at the minimum or maximum of \( D(x_1, x_2) \), the rate of change must be zero for both variables. Hence, setting these partial derivatives to zero is crucial in optimization.

###### Root-Finding System

Now that we have partial derivatives, we need to set up a root-finding system to solve for our variables. A root-finding system is a set of equations that we solve to find the 'roots' or solutions where the equations equal zero.

When we set \( \frac{\partial D}{\partial x_1} = 0 \) and \( \frac{\partial D}{\partial x_2} = 0 \), we form a system of two equations. Solving this system allows us to find the values of \( x_1 \) and \( x_2 \) that satisfy both conditions simultaneously, which are our 'roots'.

Essentially, finding these roots means we are finding the points where the function's rate of change is zero, indicating potential minimum or maximum values.

###### Optimization Problems

Optimization problems are all about finding the best value of a function under given conditions. When dealing with functions of multiple variables like \( D(x_1, x_2) \), we often aim to find points where the function reaches its highest or lowest value.

Mathematically, this involves:

- Defining the function we want to optimize.
- Calculating its partial derivatives.
- Setting these derivatives to zero to find 'critical points'.

By solving the derived system of equations, we pinpoint these critical points. Not all critical points are minimum or maximum; some could be saddle points. Analyzing second derivatives or applying other criteria helps us distinguish between them.

###### System of Equations

In our exercise, transforming the optimization problem into a system of equations is essential. A system of equations consists of multiple equations that we solve together to find solutions to the variables involved. Here, our system is:

- \( \frac{\partial D}{\partial x_1} = 0 \)
- \( \frac{\partial D}{\partial x_2} = 0 \)

Solving these equations simultaneously provides us with the values of \( x_1 \) and \( x_2 \). Once we find these values, we can check the function \( D(x_1, x_2) \) at these points to confirm if they represent a minimum or maximum. Transitioning the problem into this form makes it methodical and structured, simplifying the path to the solution.

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