MinMax Points in Multivariable Functions
Multivariable Functions have minima and maxima points, which can be found using critical points. After finding them, in order to check whether they are minima, maxima or saddle points, we can either apply the second partial derivate test which goes like:
$$ D = f_{xx}(a,b)f_yy(a,b) - f_{xy}(a,b)^2 $$
- if $D > 0$ and $f_xx(a,b) > 0$, then $(a,b)$ is a maxima
- if $D > 0$ and $f_xx(a,b) < 0$, then $(a,b)$ is a minima
- if $D < 0$ then $(a,b)$ is a saddle point.
- Otherwise, the test is inconclusive
You can also look at the Level Lines of the function and infer from there:
In a level map, if two lines intersect, that point is a saddle point