+++
date = '2026-03-02T19:00:00-06:00'
draft = false
title = 'Function of Several Variables'
layout = 'chapter'
type = 'book'
tags = 'calculus'
chapterno = 15
+++

## Max/Min Problems
-------------------

Definition:
\[
\text{Suppose } (a,b) \text{ is a point in region} R \text{ on which } f \text{ is defined and there is an open disk centered at } (a,b) \text{.}\\\\ 
\text{If: } f(x,y) \leq  f(a,b) \text{, then } f(a,b) \text{ is a local max.}\\
\text{If: } f(x,y) \geq f(a,b) \text{, then } f(a,b) \text{ is a local min.}
\]

Theorem:
\[
\text{If } f \text{ has a local max or min value at } (a,b) \text{ and the partial derivatives } f_x \text{ and } f_y \text{ exist at } (a,b) \text{, then } f_{x}(a,b) = f_{y}(a,b) = 0 \text{.}
\]

If you are at \((a,b)\) and every where you look, the value next to you is higher, you are at a local minimum position. If everywhere you look, the value is lower, you are at a local maximum position. 

Critical points are identified by finding where the partial derivatives of each variable in the multi-variate function equals zero.

Definition:
\[
\text{ A critical point in } f \text{ is located as an interior point } (a,b) \text{ if either:}\\\\
f_{x}(a,b) = f_{y}(a,b) = 0\\
f_{x} \text{ or } f{y} \text{ does not exist at } (a,b)
\]

If you are able to find the critical points of \(f\) then you can use the _Second Partial Derivative Test_ to determine the maximum and minimum values.

Theorem:
\[
\text{If the second partial derivatives of } f \text{ are continuous throughout an open disk centered at } (a,b) \text{. Let } D(x,y) = f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2} \text{.}\\\\
\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local max at } (a,b) \text{.}\\
\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local min at } (a,b) \text{.}\\
\text{If } D(a,b) < 0 \text{, there exists a saddle point at }(a,b) \text{.}\\
\text{If } D(a,b) = 0 \text{ the test is inconclusive.}
\]