Maximum and Minimum values of a function (2 vars)

In summary, the function f(x, y) = (x-y)/(2 +x2 + y2) has critical points at (1, -1) and (-1, 1). However, when checking the boundary x2 + y2 = 4, there does not seem to be any values that make the derivative of the boundary function equal to zero. This could be due to a mistake in taking the derivative of the function.
  • #1
VenaCava
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Homework Statement



Find the max and min values of the function f(x, y) = (x-y)/(2 +x2 + y2)
over the disk x2 + y2 <= 4

Homework Equations




The Attempt at a Solution




fx = 1/(2 +x2 + y2) + -2x(x-y)/(2 +x2 + y2)2
= (2- x2 + y2 +2xy)/(2 +x2 + y2)2

Should equal zero when x=1 y= -1 or x=-1 y=1

fy= (-2- x2 + y2 -2xy)/(2 +x2 + y2)2

equals zero when x=1 y=-1 or y=1 x=-1

Critical points at (1, -1) and(-1, 1)

f(-1, 1)= -1/2 f(1, -1) = 1/2

If I check the boundary

I get f(x, (4-x2)1/2) =1/6( x - (4-x2)1/2) _

If I take the derivative of that and set it to zero, I don't see any values that will let it equal 0.

f' = 1/6(1 + x/(4-x2)1/2

Shouldn't there be a max and min value on the boundary, even if it's not the abs max/min on the region.

I know I must be making a mistake somewhere, but I can't find it.

Thanks
 
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  • #2
It seems probable that your forgetting a rule that happens when you take square roots of a value. (I do it all the time. Drives me nuts, so if this is your error I understand your pain.)

As just a random example, If x2 = 9, how many different values get you to the 9?
 

What is the definition of maximum and minimum values of a function with 2 variables?

The maximum and minimum values of a function with 2 variables represent the highest and lowest points on the graph of the function. They are also known as the global maximum and minimum, as they represent the overall highest and lowest values of the function.

How do you find the maximum and minimum values of a function with 2 variables?

To find the maximum and minimum values of a function with 2 variables, you can use the method of partial derivatives. This involves taking the partial derivative of the function with respect to each variable, setting them equal to 0, and solving the resulting system of equations.

Can a function have multiple maximum or minimum values with 2 variables?

Yes, a function can have multiple maximum or minimum values with 2 variables. These are known as local maximum or minimum values, as they represent the highest or lowest points within a specific region of the graph.

What is the significance of the maximum and minimum values of a function with 2 variables?

The maximum and minimum values of a function with 2 variables are important as they help us understand the behavior of the function and its critical points. They can also help us determine the optimal values for a given set of variables.

How do maximum and minimum values of a function with 2 variables relate to optimization problems?

In optimization problems, the maximum and minimum values of a function with 2 variables represent the optimal solution. These values can help us determine the best possible outcome for a given set of variables, making them crucial in solving optimization problems.

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