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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?

Orpheus
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  • Note that the point $(1,1,6 \sqrt{3})$ satisfies g and $f(1,1,6 \sqrt{3}) > f( \sqrt{3}, \sqrt{2},1) $ – PAM1499 Aug 26 '20 at 20:12

1 Answers1

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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.

PAM1499
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