Let f be a function with two variables with continuous second order partial derivatives \(\displaystyle{f}_{{\times}},{f}_{{{y}{y}}}\ {\quad\text{and}\quad}\ {f}_{{{x}{y}}}\) at a critical point (a,b).

The necessary condition for the minimum of the function is \(\displaystyle{f}_{{{x}}}={0}\ {\quad\text{and}\quad}\ {f}_{{{y}}}={0}\) at critical point (a, b).

And the sufficient condition is,

Let,

\(\displaystyle{D}={{f}_{{\times}}{\left({a},{b}\right)}}{{f}_{{{y}{y}}}{\left({a},{b}\right)}}-{{{f}_{{{x}{y}}}^{{{2}}}}{\left({a},{b}\right)}}\)

If D>0 and \(\displaystyle{{f}_{{\times}}{\left({a},{b}\right)}}{>}{0}\), then f has a relative minimum at (a,b).

The necessary condition for the minimum of the function is \(\displaystyle{f}_{{{x}}}={0}\ {\quad\text{and}\quad}\ {f}_{{{y}}}={0}\) at critical point (a, b).

And the sufficient condition is,

Let,

\(\displaystyle{D}={{f}_{{\times}}{\left({a},{b}\right)}}{{f}_{{{y}{y}}}{\left({a},{b}\right)}}-{{{f}_{{{x}{y}}}^{{{2}}}}{\left({a},{b}\right)}}\)

If D>0 and \(\displaystyle{{f}_{{\times}}{\left({a},{b}\right)}}{>}{0}\), then f has a relative minimum at (a,b).