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.

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 $f_{yy}$