# Multivariable Calc Absolute Extrema Problem

• joemabloe
In summary, the conversation discusses finding the critical points for F(x,y)= sin(x)sin(y)sin(x+y) over the square 0\underline{}<x\underline{}<pi and 0\underline{}<y\underline{}<pi, with values for x and y from 0 to pi inclusive. The method involves taking partial derivatives in terms of x and y, setting them equal to 0, and using the product rule to combine them. The conversation also mentions finding solutions for siny=0 and sinx=0, but getting stuck.
joemabloe

## Homework Statement

F(x,y)= sin(x)sin(y)sin(x+y) over the square 0$$\underline{}<$$x$$\underline{}<$$pi and 0$$\underline{}<$$y$$\underline{}<$$pi

(The values for x and y should be from 0 to pi INCLUSIVE)

## The Attempt at a Solution

I know I need to do the partial derivatives in terms of x and y and set them equal to 0 to find the critical points, but I am having some trouble with that.

Do you know $\frac{\partial}{\partial x}\sin(x)$, $\frac{\partial}{\partial x}\sin(y)$ and $\frac{\partial}{\partial x}\sin(x+y)$? You can use the product rule to put them together- it works the same with partial derivatives. Then do the same but with $\frac{\partial}{\partial y}$.

Tomsk said:
Do you know $\frac{\partial}{\partial x}\sin(x)$, $\frac{\partial}{\partial x}\sin(y)$ and $\frac{\partial}{\partial x}\sin(x+y)$? You can use the product rule to put them together- it works the same with partial derivatives. Then do the same but with $\frac{\partial}{\partial y}$.

partial derivative in terms of x = siny[cosxsin(x+y)+sinxcos(x+y)]
you get y=0, pi because siny =0, but I don't know how to solve for the other solutions

partial derivative in terms of y = sinx[cosysin(x+y)+ sinycos(x+y)]
and you get x=0, pi because sinx=0

and then I have the same problem again and am stuck

## 1. What is a multivariable calc absolute extrema problem?

A multivariable calc absolute extrema problem is a type of mathematical problem that involves finding the maximum or minimum values of a function with multiple variables. This can be done by taking partial derivatives and using critical points to determine the absolute extrema.

## 2. How do you find the absolute extrema of a multivariable function?

To find the absolute extrema of a multivariable function, you must first take partial derivatives with respect to each variable and set them equal to zero. Then, solve for the critical points and plug them back into the original function to determine which points give the absolute maximum or minimum values.

## 3. What is the difference between absolute and relative extrema?

The absolute extrema of a function are the highest and lowest values that the function can take on over its entire domain. The relative extrema, on the other hand, are the highest and lowest values within a specific interval or region of the function.

## 4. Can a multivariable function have more than one absolute extrema?

Yes, a multivariable function can have more than one absolute extrema. This can occur when there are multiple critical points that give the same maximum or minimum value. These points are known as relative extrema.

## 5. What is the importance of finding absolute extrema in multivariable calculus?

Finding absolute extrema in multivariable calculus is important because it allows us to determine the optimal values of a function in real-world applications. For example, in economics, finding the absolute maximum profit can help a company make strategic decisions. Additionally, it is a fundamental concept in optimization, which is used in various fields such as engineering, physics, and computer science.

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