15.7 Maximum and Minimum Values MAT 3238 Vector Calculus 15.7 Maximum and Minimum Values

Homework Both written and WA HW due Thursday.

One Variable Vs Two variables

Local Maximum

Critical Numbers, Critical Points 𝑦 𝑥 𝑦=𝑓 𝑥

Fermat’s Theorem 𝑦 𝑥 𝑦=𝑓 𝑥

The Second Derivative Test

The Second Derivative Test 𝑦 𝑦=𝑓 𝑥 𝑥 𝑐 Concave up at 𝑥=𝑐

Quote When You Realize You’ve Hit Rock Bottom, There’s Only One Way To Go, And That’s Up!

Quote When You Realize You’ve Hit Rock Bottom, There’s Only One Way To Go, And That’s Up! But along which Direction?

The Second Derivative Test 𝑦 𝑦=𝑓 𝑥 𝑥 𝑐 Concave up at 𝑥=𝑐

The Second Derivative Test

The Second Derivative Test Possible LMM or Inflection Point

The Second Derivative Test

Example 1

Example 1

SageMath Codes var('x,y') plot3d(x*y*(1-x-y),(x,-1,1),(y,-1,1))

Absolute Minimum, Maximum 𝑦=𝑓(𝑥) on [𝑎,𝑏] Closed Interval Method 𝑦 𝑦=𝑓 𝑥 𝑥 𝑏 𝑎

Absolute Minimum, Maximum 𝑧=𝑓(𝑥,𝑦) on a closed and bounded region 𝐷

Example 2

Example 2

Example 2 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷

Example 2 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷

Example 2 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷

Example 2 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷

Example 2 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷

-End- For Spring 2018

Things to Do Next Year Embed a SageMath Webpage

Diagrams 𝑎 𝐼 𝑦 𝑥 𝑦=𝑓 𝑥

Diagrams 1 𝑦 𝑥 𝐿 1 𝐿 2 𝐿 3 𝐿 4 𝐷