- #1

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## Homework Statement

Find max/min of x^2+y^2+z^2 given x^4+y^4+z^4=3

## Homework Equations

Use of gradient vectors related by LaGrange Multiplier

## The Attempt at a Solution

[tex]\begin{gathered}

f\left( {x,y,z} \right) = {x^2} + {y^2} + {z^2};g\left( {x,y,z} \right) = {x^4} + {y^4} + {z^4} - 3 = 0 \\

\vec \nabla f = \left\langle {2x,2y,2z} \right\rangle ;\vec \nabla g = \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\

\left\langle {2x,2y,2z} \right\rangle = \lambda \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\

2{x^2} = 2{y^2} = 2{z^2} \to x = \pm y = \pm z \\

3{x^4} - 3 = 0 \to {x^4} = 1 \to x = \pm 1 \to y = \pm 1,z = \pm 1 \\

\max = f\left( {1,1,1} \right) = f\left( {1,1, - 1} \right) = f\left( {1, - 1,1} \right) = f\left( {1, - 1, - 1} \right) = \\

f\left( { - 1,1,1} \right) = f\left( { - 1,1, - 1} \right) = f\left( { - 1, - 1,1} \right) = f\left( { - 1, - 1, - 1} \right) = 3 \\

\end{gathered}[/tex]

So I found the maximum but does the minimum exist?