Question

For a function of two variables, describe relative minimum.

Derivatives
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asked 2021-02-17
For a function of two variables, describe relative minimum.

Answers (1)

2021-02-19
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).
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