# Chapter 22: Extreme Values

### Extreme Values

We have the detailed study of the structure of graphs of functions. Flow are they shaped? What are their key properties?

Definition → Let a function ﻿$y=f(x)$﻿ be defined on the domain ﻿$D$﻿.

﻿$x=c$﻿ is an ABSOLUTE MAXIMUM if →

﻿$f(x) \leq f(c)$﻿ for all ﻿$x$﻿ in ﻿$D$﻿

﻿$x=c$﻿ is an ABSOLUTE MINIMUM if

﻿$f(c) \leq f(x)$﻿ for all ﻿$x$﻿ in ﻿$D$﻿

We say there are EXTREME VALUES of ﻿$f(x)$﻿

#### EXAMPLE

Find the global min of ﻿$y=x^{2}$﻿ on ﻿$D =(-\infty,\infty)$﻿

We know ﻿$x^{2} \geq 0$﻿ thus ﻿$x=0$﻿ is the minimum

#### Example

If possible find the global min/max of ﻿$y=x^{2}$﻿ on the domain ﻿$D=[1,2]$﻿

We have ﻿$x=1$﻿ is a minimum.However, ﻿$x=2$﻿ is not in the domain and thus, there is no max.

Discuss → Fill in the table with domains for the function

﻿$y=sin(x)$﻿

Definition → A function ﻿$y=f(x)$﻿ has a LOCAL MAX ﻿$f(x)\leq f(c)$﻿ for all ﻿$x$﻿ near ﻿$c$﻿. ﻿$f(x)$﻿ has a LOCAL MIN at ﻿$x=c$﻿

f(c) ≤ f(x) for all x near c

"Near" - Means there is a small number t so that -

﻿$\mid x-c\mid < t \Rightarrow f(c)\leq f(x)$﻿ for minima

OR

﻿$\mid x-c\mid < t \Rightarrow f(x)\leq f(c)$﻿ for maxima

Discuss → Label the local/global min/max of -

Fact → (The Extreme Value Theorem)

Any continuous function defined on a closed interval ﻿$[a,b]$﻿ has a global maximum and a global minimum.

This fact like the IVT, is not easy to prove. It is tricky and topological.

Fact → If ﻿$x=c$﻿ is a local min or max and ﻿$f'(c)$﻿ is defined then ﻿$f'(c)=0$﻿

Example - Find the local min/max of ﻿$f(x)=x^{3}-3x$﻿

# Find the derivative of ﻿$f(x)$﻿

﻿$f'(x)=3x^{2}-3$﻿

# Find the roots of the derivative

﻿$f'(x)=3x^{2}-3=0 \Rightarrow x^{2}-1=0 \Rightarrow \pm 1$﻿

Thus, ﻿$x=-1$﻿ and ﻿$x=1$﻿ may be local max/min

Definition → A point where ﻿$f'(x) =0$﻿ or is undefined is a CRITICAL POINT of ﻿$y=f(x)$﻿

Algorithm Find the global min/max of ﻿$f(x)$﻿ on a finite closed intervals

1. Find the critical points of ﻿$f(x)$﻿
2. Evaluate ﻿$f(x)$﻿ at all endpoints and critical points
3. Take the largest and smallest values

#### Example

Find the global min/max of ﻿$y=\mid x\mid$﻿ on ﻿$[-2,5]$﻿

1. ﻿$f'(x)=0$﻿ or undef ﻿$\Rightarrow x=0$﻿
2. ﻿$f(-2)=\mid -2\mid =2$﻿

﻿$f(0)=\mid 0\mid =0$﻿

﻿$f(5)=\mid 5\mid =5$﻿

3. Thus

﻿$x=0$﻿ is the global min

﻿$x=5$﻿ is the global max.

#### Example

Find the global min/max of ﻿$f(x)=xe^{-x}$﻿ on ﻿$[-1,1]$﻿

1. ﻿$f'(x)=e^{-x}+xe^{-x}(-1)$﻿

﻿$=e^{-x}-xe^{-x}$﻿

﻿$=(1-x)e^{-x}$﻿

﻿$f'(x)=0 \Rightarrow x=1$﻿

2. ﻿$f(-1)=1e^{1}=-e$﻿

﻿$f(1)=1\cdot e^{-1}=\frac{1}{e}$﻿

3. Thus,

﻿$x=-1$﻿ is the min

﻿$x=1$﻿ is the max

#### Example

Find all the critical points of ﻿$f(x)=6x^{2}-x^{3}$﻿

﻿$f'(x)=12x-3x^{2}=0$﻿

﻿$=3x(4-x)$﻿

Thus, ﻿$x=0$﻿ OR ﻿$x=4$﻿

#### Example

Find all critical points of ﻿$f(x)=in(x+1)-arctan(x)$﻿

﻿$f'(x)=\frac{1}{x+1} -\frac{1}{x^{2}+1}$﻿

﻿$=\frac{(x^{2}+1)-(x+1)}{(x^{2}+1)}$﻿

﻿$=\frac{x^{2}-x}{(x+1)(x^{2}+1)}$﻿

﻿$=\frac{x(x-1)}{(x+1)(x^{2}+1)}$﻿

Thus, ﻿$f(x)$﻿ has critical points at -

﻿$x=1 \Rightarrow f'(x)$﻿ undef

﻿$x=0 \Rightarrow f'(x)=0$﻿

﻿$x=1 \Rightarrow f'(x)=0$﻿