## calculus@internet

Area 6.5.1

Maxima and Minima from Contour Maps

## Local Maxima and Minima of F(x,y)

The input values (a,b) where a function F(x,y) has a local minimum output
value or a local maximum output value is called a critical point.
Critical points live in the domain of the function F(x,y). So do the contours
of F(x,y). Therefore, it seems reasonable to look for maxima and minima in
the contour diagrams.

The general process in finding local maxima and minima from contours is to graph the
contour diagram of a function F(x,y), and then "look for the maxima and minima"
by looking for concentric circular-type regions culminating in a point -
a critical point. Here we see the comparison between the 2D contour graph and the
3D surface graph, and its maxima and minima for this region.

The advantage of the contour graph is that you can spot more easily the input
coordinate values than you can with a 3D graph. Here is an exercise for
you to work.
## Exercise #1: Finding Critical Values from Contour Diagrams

## Exercise #2: Finding Critical Values from Contour Diagrams

## Exercise #3: Finding Critical Values from Contour Diagrams

## Turn-in Alert!

*You will turn-in this notebook to your instructor's mailbox.*

## What is the 2D Gradient Doing at a Critical Point?

I am not going to tell you quite yet. Let's let you figure this one out. Our
goal will be to complete the following theorem statements:
#1. If (a,b) is a critical point for the function F(x,y), then the
gradient of F evaluated at (a,b) is ...

#2. If (a,b) is a critical point for the function F(x,y), then the
partial derivatives of F evaluated at (a,b) are ...

#3. If (a,b) is a critical point for the function F(x,y), then the
tangent plane at (a,b) is ...

## Turn-in Alert!

*You will turn-in this notebook to your instructor's mailbox.*
## Vector Field Generator

Here is a Theorist Notebook that has a 2D Vector Field Generator in it.
You may use this notebook to generate either an arbitrary 2D Vector Field,
or the Gradient Field of a 3D Surface (i.e., THE DERIVATIVE of the 3D Surface).
## Lagrange Multiplier Notebook

This notebook will allow you to graph a function F(x,y) and the constraining
cylindricized curve automatically.

## Road Map

`calculus`@`internet` lives at::
` http://mac205.sjdccd.cc.ca.us/cai-home.html`

© 1995,
Robert R. Curtis and Phil R. Smith.
Printed version: © 1995, The Finley Press, Stockton, CA, USA

`curtis@ms.sjdccd.cc.ca.us` ;
`prsmith@ucdavis.edu`