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## Homework Statement

Lagrange multipliers to find the maximum and minimum values of f(x,y) = 4x^3 + y^2 subject to the constraint 2x^2 + y^2 = 1. Find points of these extremum.

## Homework Equations

## The Attempt at a Solution

g(x,y)= 2x^2 + y^2 - 1

f(x,y)= 4x^3 + y^2

Gradient F= 12x^2i + 24yj

Gradient G= 4xi + 2yj

Gradient = 4x^3 + y^2 - λ(2x^2 + y^2 - 1)

= [12x^2- λ4x, 2y - 2λy, -2x^2 - y^2 - 1]

12x^2 - λ4x = 0

3x = λ

2y - 2λy = 0

λ= 1

x = 1/3 (Is this correct?)

fx (x,y) = 12x^2 x=0

fy (x,y) = 2y y=0

No minima or maxima is my conclusion but I'm very sure it's wrong. Also is the 3-D representation somewhat like a paraboloid with a ellipsoid constraint?

Thanks!