Answer
Maximum:$\sqrt 3$ and minimum: $1$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$
This yields $\nabla f=\lt 2x,2y,2z \gt$ and $\lambda \nabla g=\lambda \lt 4x^3,4y^3,4z^3\gt$
Using the constraint condition $x^4+y^4+z^4=1$ we get, $2x=\lambda 4x^3,2y=\lambda 4y^3,2z=\lambda 4z^3$
After solving, we get $x=y=z$
Since, $g(x,y)=x^4+y^4+z^4=1$ gives $x=\sqrt[4] {\dfrac{1}{3}}$
Thus, $x=y=z=\sqrt[4] {\dfrac{1}{3}}$
Also, we have possible points:$x=y=0,z=\pm 1$
Hence, maximum:$\sqrt 3$ and minimum: $1$
