Finding extreme values of a multivariable function

The general idea is analogous to single variable functions. Whether we are discussing the function's domain or a subdomain of it, we have to use derivatives to analyse how the function behaves to know whether a point is a maximum or a minimum. For two variables the idea is the same as for one variable, in a certain interval the function ca be constant, crescent or decrescent. For three and more variables we lose the function's graph, but the algebra is the same. In case the function is strictly crescent or strictly decrescent, there isn't a maximum or a minimum unless we define a closed interval.

Let [math]\displaystyle{ P = (x,y,z,...,n) }[/math] be a point of many coordinates and [math]\displaystyle{ f }[/math] a multivariable function with a domain [math]\displaystyle{ D }[/math]. [math]\displaystyle{ P_0 \in D }[/math] is a point of maximum of [math]\displaystyle{ f }[/math] if [math]\displaystyle{ f(P) \leq f(P_0) }[/math] for all [math]\displaystyle{ P \in D }[/math]; and [math]\displaystyle{ P_0 }[/math] is a point minimum if [math]\displaystyle{ f(P) \geq f(P_0) }[/math] for all [math]\displaystyle{ P \in D }[/math].