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## Homework Statement

Find extreme values for

[tex]f(x, y, z)=x+y+z[/tex]

subject to:

[tex]x^4+y^4+z^4=c>0[/tex]

Therefore, since c>0, exclude the origin.

Let [tex]L(x, y, z, \lambda{)}=x+y+z+\lambda{(}x^4+y^4+z^4-c)[/tex]

and thus

[tex]L_1(x, y, z, \lambda{)}=1+4x^3\lambda[/tex]

[tex]L_2(x, y, z, \lambda{)}=1+4y^3\lambda[/tex]

[tex]L_3(x, y, z, \lambda{)}=1+4z^3\lambda[/tex]

[tex]L_4(x, y, z, \lambda{)}=x^4+y^4+z^4-c[/tex]

with solutions

[tex]x=y=z=(-1/4\lambda{)}^{1/3}[/tex]

and

[tex]0=((-1/4\lambda{)}^{1/3})^4+((-1/4\lambda{)}^{1/3})^4+((-1/4\lambda{)}^{1/3})^4-c=3((-1/4\lambda{)}^{1/3})^4-c[/tex]

[tex]c=-1/(4c^{3/4})[/tex]

and since c>0, the question has no solutions.