- #1
DougD720
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Homework Statement
Find the extrema of f(x,y)=x-y ; subject to x2-y2=2
Homework Equations
[tex]\nabla[/tex]f=[tex]\lambda[/tex][tex]\nabla[/tex]g
The Attempt at a Solution
[tex]\nabla[/tex]f=(1,-1)
[tex]\nabla[/tex]g=(2x, -2x)
(1,-1)=[tex]\lambda[/tex](2x, -2x)
1 = [tex]\lambda[/tex](2x) -> [tex]\lambda[/tex]=[tex]\frac{1}{2x}[/tex]
-1 = [tex]\lambda[/tex](-2y) -> [tex]\lambda[/tex]=[tex]\frac{1}{2y}[/tex]
Which means x = y , but it has to satisfy x2-y2=2 and if x=y then it cannot satisfy this meaning there are no extrema for this set of equations.
Am i right? I tried working it out with other methods but it just keeps not working, however, i plotted the two equations in 3D on Maple and they do intersect so shouldn't there be extrema? Or is the fact that x-y is a plane parallel to the xy-axis mean that all points are extrema?
We never did a problem like this in class, one with no apparent solution, so I'm confused a bit here.
And i just did another problem where I'm coming up with a solution that doesn't satisfy one of the constraints... ugh... what am i doing wrong?
Thanks for the help!