# Find the maximum and minimum values of the function (Answers included).

• s3a
In summary, the problem asks to find the maximum and minimum values of the given function on a disk, using the Lagrange multiplier method and partial differentiation. The Lagrange multiplier method requires adding or subtracting a term with λ based on whether you want to find the maximum or minimum. The sign of λ does not matter in equality constrained problems. The method can also be used to find values on the boundary by setting the partial derivatives equal to 0.
s3a

## Homework Statement

Find the maximum and minimum values of the function
f(x,y) = 2x^2 - 28xy + 2y^2 - 1 on the disk x^2 + y^2 <= 1.

(a) Find the maximum.
(b) Find the minimum.

## Homework Equations

Lagrange Multiplier method and partial differentiation.

## The Attempt at a Solution

My work is attached as MyWork.jpg. If anything is unclear, tell me and I will restate it (or whatever it is you request). In the image, I forgot to plug in x and y in f(x,y) to get f(sqrt(1/2), sqrt(1/2). This yields the correct answer for the minimum: 2*(sqrt(1/2))^2 - 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1 whereas the correct answer for the maximum is: 2*(sqrt(1/2))^2 + 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1. So, my ultimate question is, what are the reasons for choosing positive or negative square roots for x and/or y?

#### Attachments

• MyWork.jpg
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s3a said:

## Homework Statement

Find the maximum and minimum values of the function
f(x,y) = 2x^2 - 28xy + 2y^2 - 1 on the disk x^2 + y^2 <= 1.

(a) Find the maximum.
(b) Find the minimum.

## Homework Equations

Lagrange Multiplier method and partial differentiation.

## The Attempt at a Solution

My work is attached as MyWork.jpg. If anything is unclear, tell me and I will restate it (or whatever it is you request). In the image, I forgot to plug in x and y in f(x,y) to get f(sqrt(1/2), sqrt(1/2). This yields the correct answer for the minimum: 2*(sqrt(1/2))^2 - 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1 whereas the correct answer for the maximum is: 2*(sqrt(1/2))^2 + 28*sqrt(1/2)*sqrt(1/2) + 2(sqrt(1/2))^2 - 1. So, my ultimate question is, what are the reasons for choosing positive or negative square roots for x and/or y?

In the problem max f(x) s.t. g(x) <= 0, if you want the Max you should use L = f - λg, with λ ≥ 0; if you want the Min you should use L = f + λg, with λ >= 0 (or use f - λg with λ ≤ 0).

How can you remember this? The easiest way is to form L so that it is more favourable than f for feasible points; that is, for feasible points, L should be ≥ f in a max problem and should be ≤ f in a min problem. So, if the constraint is g ≤ 0 we get something ≥ f in a max problem by subtracting g, that is, by using L = f - λg with λ ≥ 0.

Note that for an equality constrained problem, with constraint g = 0, the sign of λ is not determined, and it does not matter whether you write L = f + λg or L = f - λg.

So, in your analysis you need to look at the sign of your λ.

RGV

The Lagrange multiplier method requires that you have "=" in the constraint. You can use that to determine any max or min on the boundary. Just set the partial derivatives equal to 0 in the interior of the disc.

## What is the function used to find the maximum and minimum values?

The function used to find the maximum and minimum values is called "max" and "min" respectively. These functions are commonly used in mathematical and scientific computations to determine the highest and lowest values in a set of data.

## What are the inputs required to find the maximum and minimum values?

The inputs required to find the maximum and minimum values of a function are the independent variables or inputs of the function. These can be numerical values, equations, or expressions that are used to calculate the output or dependent variable of the function.

## How do you interpret the maximum and minimum values of a function?

The maximum value of a function is the highest point on the graph of the function, while the minimum value is the lowest point. These values represent the extreme or optimal values of the function and can be used to analyze and optimize real-world situations.

## What are the methods used to find the maximum and minimum values?

There are various methods used to find the maximum and minimum values of a function, including graphical methods, algebraic methods, and calculus-based methods. These methods involve analyzing the behavior and properties of the function to determine its highest and lowest points.

## Can the maximum and minimum values of a function change?

Yes, the maximum and minimum values of a function can change as the inputs or independent variables of the function change. This is because the shape and behavior of a function can vary depending on its inputs, resulting in different maximum and minimum values.

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