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## Homework Statement

Use the method of Lagrange multipliers to ﬁnd the maximum and minimum values of the function

f(x, y) = x + y

^{2}

subject to the constraint g(x,y) = 2x

^{2}+ y

^{2}- 1

## Homework Equations

none

## The Attempt at a Solution

We need to find [itex]\nabla[/itex]f = λ[itex]\nabla[/itex]g

Hence,

[itex]\nabla[/itex]f

_{x}- λ[itex]\nabla[/itex]g

_{x}= 0

Which becomes, 1 - λ(4x) = 0

[itex]\nabla[/itex]f

_{y}- λ[itex]\nabla[/itex]g

_{y}= 0

Which becomes, 2y - λ = 0

-------------------------

Now we have: x = 1/4λ and y = λ/2

I assume I am right in now subbing x and y into the constraint...

To give us: 2/16λ

^{2}+ λ

^{2}/4 - 1 = 0

It seems a bit messy considering this is an elementary part of my homework? Have I gone wrong somewhere?

Regards as always