Ask question
Question

# For a function of two variables, describe relative minimum.

Derivatives
ANSWERED
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).

expert advice

...