We have the function $g(x,y,z)=xy^2z$ and the hypersurface $f(x,y,z) = x^4 + y^4 + z^4 = 1$. To find the global max and min (which result in $\frac{1}{2\sqrt{2}}$ and $-\frac{1}{2\sqrt{2}}$) I used Lagrange Multipliers:
(gradient of g) = (lambda)(gradient of f)
$(y^2z,2xyz,xy^2)= λ(4x^3,4y^3,4z^3)$
$2xyz = 4y^3$ / $y^2z = 4x^3$ --> $2x^4=y^4$ and $2z^4=y^4$
So, $y=\frac{1}{\sqrt{\sqrt{2}}},-\frac{1}{\sqrt{\sqrt{2}}}$
$x=\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$
$y=\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$
Therefore, the global max of $xy^2z = \frac{1}{2\sqrt{2}}$ and the global min is $-\frac{1}{2\sqrt{2}}$
How do I find the points $(x,y,z)$ that are on $g$ and $f$ and are not global maxima/ minima. In other words, how do I find the local extrema?