- #1
ChrisPls
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Homework Statement
I need to find the extrema of [itex]f(x,y) = 3x^{2} + y^{2}[/itex] given the constraint [itex]x^{2} + y^{2} = 1[/itex]
Homework Equations
I'm not sure what goes here. I've been trying to solve it with this:
∇f(x,y) = λ∇g(x,y)
The Attempt at a Solution
[itex]f(x,y) = 3x^{2} + y^{2}[/itex]
[itex]g(x,y) = x^{2} + y^{2} = 1[/itex]
[itex]∇f(x,y) = <6x, 2y>[/itex]
[itex]∇g(x,y) = <2x, 2y>[/itex]
With the previous equation, we get a system of equations:
[itex]6x = 2xλ[/itex]
[itex]2y = 2yλ[/itex]
and the constraint
[itex]x^{2} + y^{2} = 1[/itex]
Solving for x and y, I get x=y=0. But that fails the final constraint. I've used the above procedure for other Lagrange problems without much issue, but this one's stumping me.
So I tried finding a solution on Wolfram: Maxes at (-1, 0) and (1, 0), and Mins at (0, -1) and (0, 1). Okay, nice and simple looking solutions. Right?
But I just can't figure this out. What am I missing?
Thanks