Question

Please how do we distinguish or know when it is a minimum or maximum or saddle point in an extrema of two or more variables(multivariable) function.

Answer 1

To  distinguish between minimum or maximum or saddle points one can use the second partial derivative test.

Let's define the  Hessian matrix:

$H(x,y) = \begin{pmatrix}f_{xx}(x,y) &f_{xy}(x,y)\\f_{yx}(x,y) &f_{yy}(x,y)\end{pmatrix}$ ,

with partial derivatives of the function $f(x,y)$ .

Let $D(x,y)$ be the determinate of $H(x,y)$ :

$D(x,y)=f_{xx}(x,y)f_{yy}(x,y) - \left( f_{xy}(x,y) \right)^2$ .

If $x = a, y = b$ - is critical point of function, then we have the following distinguish rule:

1. If $D(a,b)>0$ and $f_{xx}(a,b)>0$ then $(a,b)$ is a local minimum of $f(x,y)$ .
2. If $D(a,b)>0$ and $f_{xx}(a,b)<0$ then $(a,b)$ is a local maximum of $f(x,y)$ .
3. If $D(a,b)<0$ then $(a,b)$ is a saddle point of $f(x,y)$ .
4. If $D(a,b)=0$ then the second derivative test is inconclusive, and one should use Higher-order derivative test in order to classify the critical point.