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