- #1

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## Homework Statement

Find the max and min values of the function f(x, y) = (x-y)/(2 +x

^{2}+ y

^{2})

over the disk x

^{2}+ y

^{2}<= 4

## Homework Equations

## The Attempt at a Solution

fx = 1/(2 +x

^{2}+ y

^{2}) + -2x(x-y)/(2 +x

^{2}+ y

^{2})

^{2}

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

^{2}+ y

^{2})

^{2}

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

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

^{2}+ y

^{2})

^{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-x

^{2})

^{1/2}) =1/6( x - (4-x

^{2})

^{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-x

^{2})

^{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