# Finding Rel. Max, Min, and Saddle Points on Levelset

• Loppyfoot
In summary, the diagram above shows a collection of level sets for a function, with the outer-most level being at the lowest height. Points A-E on the diagram represent critical points, with point B being a relative maximum. The nature of points C, D, and A is not specified.
Loppyfoot

## Homework Statement

The diagram above represents a collection of level sets for a certain function, where the outer-most level is at the lowest height.

What are points A-E? relative min, relative max, saddle point, or not a critical point

## The Attempt at a Solution

I have tried multiple times to get this question right, but I keep failing.

I know that point B is a relative Maximum.
What would C, D , and A be?

#### Attachments

• cfe9ed6c-b289-4978-a8f9-0188f7166ee5.gif
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Does Anyone have any Ideas? I don't think its that difficult of a question for some, but its just a little bit confusing.

## 1. What is the purpose of finding relative maximum, minimum, and saddle points on a levelset?

Finding relative maximum, minimum, and saddle points on a levelset is important in optimization problems to determine the optimal values of a function. It also helps in identifying critical points and understanding the behavior of the function.

## 2. How do you find the relative maximum and minimum points on a levelset?

To find the relative maximum and minimum points on a levelset, you need to take the partial derivatives of the function and set them equal to zero. Then, solve the resulting system of equations to find the critical points. From there, you can use the second derivative test to determine whether the critical points are relative maximum or minimum points.

## 3. What is a saddle point on a levelset?

A saddle point on a levelset is a point where the function has a critical point but is neither a relative maximum nor a relative minimum. At a saddle point, the function changes from increasing to decreasing or vice versa in at least one direction.

## 4. Can there be more than one relative maximum or minimum point on a levelset?

Yes, there can be multiple relative maximum or minimum points on a levelset. This is because a function can have multiple critical points where the first derivative is equal to zero. However, each critical point may not necessarily be a relative maximum or minimum point.

## 5. How can finding relative maximum, minimum, and saddle points help in real-world applications?

In real-world applications, finding relative maximum, minimum, and saddle points can help in optimizing processes, such as maximizing profits or minimizing costs. It can also aid in understanding the behavior of a system and predicting its future behavior based on the identified critical points.

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