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Optimization using Lagrange multipliers

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1. Homework Statement [/b]

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

2. Homework Equations



3. 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= [tex]\pm[/tex]y
x^4+x^4=2
x=y= [tex]\pm1[/tex]
max= 1+1=2 @ [tex]\left(1,1\right)[/tex] and [tex]\left(-1,-1\right)[/tex]

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



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
 
Last edited:

HallsofIvy

Science Advisor
Homework Helper
41,683
865
Re: optimization

1. Homework Statement [/b]

f[tex]\left(x,y\right)[/tex] = x^2 +y^2
g[tex]\left(x,y\right)[/tex] = 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?

2. Homework Equations



3. 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= [tex]\pm[/tex]y
x^4+x^4=2
x=y= [tex]\pm1[/tex]
max= 1+1=2 @ [tex]\left(1,1\right)[/tex] and [tex]\left(-1,-1\right)[/tex]

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



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
 
13
0
Re: optimization

yes the constraint function is incorrect, i ommitted =2
 

HallsofIvy

Science Advisor
Homework Helper
41,683
865
Re: optimization

You have found that y2= x2 and you know that [itex]x^4+ y^4= 2[/itex]. That tells you that [itex]x= \pm 1[/itex] and [itex]x= \pm 1[/itex]. 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.
 
13
0
Re: optimization

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.
 

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