Optimization using Lagrange multipliers

[tinkus]

1. Homework Statement [/b]

f\left(x,y\right) = x^2 +y^2
g\left(x,y\right) = x^4+y^4 = 2
Find the maximum and minimum using Lagrange multiplier

Homework Equations

The Attempt at a Solution

grad f = 2xi +2yj
grad g= 4x^3i + 4y^3j

grad f= λ grad g
2x=4x^3λ and 2y= 4y^3λ
2x^2 = 2y^2
x^2=y^2
x= \pmy
x^4+x^4=2
x=y= \pm1
max= 1+1=2 @ \left(1,1\right) and \left(-1,-1\right)

I don't know how to find the min and not sure about the max above

 

[HallsofIvy]

1. Homework Statement [/b]

f\left(x,y\right) = x^2 +y^2
g\left(x,y\right) = x^4+y^4
Find the maximum and minimum using Lagrange multiplier

My recomendation is that you go back and read the problem carefully! What you have written here makes no sense. Usually you use Lagrange multiplier method maximize or minimize a function subject to some constraint. You have two functions with no constraint. Is one of those, either f or g, supposed to be equal to a number?

Homework Equations

The Attempt at a Solution

grad f = 2xi +2yj
grad g= 4x^3i + 4y^3j

grad f= λ grad g
2x=4x^3λ and 2y= 4y^3λ
2x^2 = 2y^2
x^2=y^2
x= \pmy
x^4+x^4=2
x=y= \pm1
max= 1+1=2 @ \left(1,1\right) and \left(-1,-1\right)

I don't know how to find the min and not sure about the max above

 

[tinkus]

yes the constraint function is incorrect, i ommitted =2

 

[HallsofIvy]

You have found that y²= x² and you know that x^4+ y^4= 2. That tells you that x= \pm 1 and x= \pm 1. That gives you four possible points: (1, 1), (-1, -1), (1, -1), and (-1, 1). You might want to consider whether there are both maximum and mimimum values.

 

[tinkus]

the points all equal 2(max). i still need to find the min which according to the answer key is sqrt2, I don't know how to get that.