Lagrange Multipliers Maxima or Minima? If I evaluate a surface $f$$($$x$$)$, that is subjected to a constraint $g$$($$x$$)$, for it's maximum and minimum values using Lagrange Multipliers then how do I know that the solution that is found is maximum or minimum.
For example $f$$($$x$$)$= $x^2$+$y^2$+$z^2$
and $g$$($$x$$)$=$x^3$$y^2$$z$=  $6$$\sqrt{3}$
The solution using Lagrange multiplier is ($\sqrt{3}$,$\sqrt{2}$,$1$)
But is this point a maxima or minima?

Since the point $(1,1,6 \sqrt{3})$ satisfies g and $f(1,1,6 \sqrt{3}) > f( \sqrt{3}, \sqrt{2},1) $ the point $( \sqrt{3}, \sqrt{2},1)$ must be a minima.
