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## Homework Statement

Let g(x,y) = x

^{2}+ 4y

^{2}. What is the maximum value of F(x,y) = ln(x

^{4}y

^{5}) on the intersection of the level set g(x,y) = 9 with the quadrant {(x,y): x>0 and y>0}

## The Attempt at a Solution

It seems I'm having a lot of difficulty with Lagrange multipliers, but here I go.

F

_{x}= 4/x = g

_{x}= 2x * λ

F

_{y}= 5/y = g

_{y}= 8y * λ

Then clearly, x

^{2}= 2/λ

and, y

^{2}= 5/(8λ)

Plugging these into the constraint gives,

g(2/λ,5/(8λ)) = 4/λ

^{2}+ 4 * (25/(64 * λ

^{2})) = 9.

Attempting to solve for λ gives,

(4/λ

^{2}) (1 + 25/64) = 9

4 + 25/16 = 9λ

^{2}

λ = ± √(4/9 + 25/144). But I reject the negative because it will yield answers outside the boundary

x = sqrt(2/λ) = sqrt(2/(sqrt((4/9 + 25/144))) ≈ 1.594990568

y = sqrt(5/(8λ) = sqrt(5/(8sqrt((4/9 + 25/144))) ≈ 0.89162683338

So I get an extremum at F(1.594990568,0.89162683338), whatever that is...

Pretty sure this isn't right. Ideas?

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