Multivariable Max and min value problem.

In summary, to find the local max, min, and saddle points of the function f(x,y)=sin(x)sin(y) on the interval -pi<x<pi and -pi<y<pi, we must first find the critical points where both partial derivatives, fx and fy, are equal to 0. These critical points occur at (0,0), (pi/2,pi/2), and (-pi/2,-pi/2).
  • #1
yaho8888
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0

Homework Statement


find local max min and saddle point.
f(x,y)= sin(x)sin(y), -pi<x<pi, -pi<y<pi

Homework Equations


none

The Attempt at a Solution


fx = cos(x)sin(y)
fy= sin(x)cos(y)

now how do I get the critical points, I know how to get max min and saddle point, but I don't know how to get critical points from this equation. when fx fy = 0, we got the critical point, I know there is (0,0), how do I find the others. I got another points (pi/2,pi/2), (-pi/2,-pi/2). is there more?
 
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  • #2
What do you mean by "fxfy= 0"? The product? A "critical point" is defined as a point where the function is not differentiable or where the partial derivatives are equal to 0. Since this function is obviously differentiable everywhere, its critical points are where cos(x)sin(y)= 0 and sin(x)cos(y)= 0. Since sin(x) and cos(x) can't be 0 at the same x, you must have either sin(x)=0 and sin(y)= 0 or cos(x)= 0 and cos(y)= 0. Where is sine 0?
 
  • #3
fx is the derivative of the function respect to x
fy ........... y
Where is sine 0?
at zero sin is zero
 
  • #4
yaho8888 said:
fx is the derivative of the function respect to x
fy ........... y
Where is sine 0?
at zero sin is zero

Your critical points will occur at points where both partials are zero. on the given intervals, what values of x and y will make both fx and fy zero?
 
  • #5
solved!
 

1. What is a multivariable max and min value problem?

A multivariable max and min value problem is a mathematical problem in which the goal is to find the maximum and minimum values of a function with multiple variables. This involves finding the values of the variables that will result in the highest and lowest outputs for the function.

2. What is the difference between a single-variable and a multivariable max and min value problem?

In a single-variable max and min value problem, there is only one independent variable. This means that the function only depends on one input. In a multivariable max and min value problem, there are multiple independent variables, and the function depends on more than one input.

3. How do you solve a multivariable max and min value problem?

To solve a multivariable max and min value problem, you need to use calculus techniques such as partial derivatives and critical point analysis. This involves finding the partial derivatives of the function with respect to each variable, setting them equal to zero, and solving for the variables.

4. What are some real-world applications of multivariable max and min value problems?

Multivariable max and min value problems are commonly used in fields such as economics, engineering, and physics to optimize various processes. For example, in economics, these problems can be used to determine the combination of inputs that will result in the maximum profit for a company.

5. Can multivariable max and min value problems have more than one solution?

Yes, multivariable max and min value problems can have multiple solutions. These solutions can represent different combinations of variables that result in the same maximum or minimum value for the function. It is important to carefully analyze the solutions to determine which one is the most optimal for the given problem.

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