Multivariate Function

The range of a multivariate function is always a subet of $\mathbb{R}$, the domain will be a subset of ${\mathbb{R}}^n$.

We can view the input as a vector. This is where Calculus and Calculus and Linear Algebra

This kind of function is referred as a Scalar Field.

Critical Point (Multivariate)

For a multivariate function, we define a critical point as a point at which either both $f_x$ and $f_y$ are zero, or else one of them is undefined.

title: Second-Derivative Test for Local Extrema
Suppose $P_0$ is a critical point of a function $f(x,y)$, and suppose that the second-order partial derivatives of $f$ are continuous in some neighbourhood of $P_0$.
 
Let $D(x,y) = f_{xx}f_{yy} - (f_{xy})^2$
- If $D(P_0) > 0$, then $f$ has an extremum at $P_0$. 
	- If $f_{xx}(P_0) < 0$* then this extremum is a maximum, whereas if $f_{xx}(P_0) > 0$ then it is a minimum.
- If $D(P_0) < 0$, then $f$ does not have an extremum at $P_0$ (that is, it has a saddle point instead).
- If $D(P_0) = 0$, the test gives no conclusion.
 
*we can also use $f_{yy}$