MAT 1234Calculus I

Section 3.1Maximum and Minimum

Values

http://myhome.spu.edu/lauw

Next WebAssign 3.1 Quiz– 2.8

1 Minute… You can learn all the important concepts

in 1 minute.

1 Minute… High/low points – most of them are at points

with horizontal tangent

1 Minute… High/low points – most of them are at points

with horizontal tangent.

Highest/lowest points – at points with horizontal tangent or endpoints

1 Minute… You can learn all the important concepts

in 1 minute. We are going to develop the theory

carefully so that it works for all the functions that we are interested in.

There are a few definitions…

Preview Definitions

• absolute max/min• local max/min• critical number

Theorems• Extreme Value Theorem• Fermat’s Theorem

The Closed Interval Method

Max/Min We are interested in max/min values

• Minimize the production cost• Maximize the profit• Maximize the power output

Definition (Absolute Max) has an absolute maximum at on if for all in ( =Domain of )

c

D

Definition (Absolute Min) has an absolute maximum at on if for all in ( =Domain of )

cD

Definition The absolute maximum and minimum

values of are called the extreme values of .

x

Example 1y

Absolute max.

Absolute min.

Definition (Local Max/Min) has an local maximum at if for all in some open interval containing .

has an local minimum at if for all in some open interval containing .

x

Example 1y

Local max.

Local min.

Q&A An end point is not a local max/min,

why?

The Extreme Value Theorem If is continuous on a closed interval ,

then attains an absolute max value and an absolute min value at some numbers c and d in .

The Extreme Value Theorem If is continuous on a closed interval ,

then attains an absolute max value and an absolute min value at some numbers c and d in .

No guarantee of absolute max/min if one of the 2 conditions are missing.

Q&A Give 2 examples of functions on an

interval that do not have absolute max value.

Example 2 (No abs. max/min) is not continuous on

𝑥

𝑦

𝑏𝑎

𝑦= 𝑓 (𝑥 )

𝑐

Example 2 (No abs. max/min) The interval is not closed

𝑥

𝑦

𝑏𝑎

𝑦= 𝑓 (𝑥 )

How to find Absolute Max./Min.? The Extreme Value Theorem guarantee

of absolute max/min if is continuous on a closed interval .

Next: How to find them?

Fermat’s Theorem If has a local maximum or minimum at ,

and if exists, then

c x

y

Q&A: T or F The converse of the theorem:If, then has a local maximum or minimum at .

Definition (Critical Number) A critical number of a function is a

number c in the domain of such that either or does not exist.

Critical Number (Translation) Critical numbers give all the potential

local max/min values

( ) 0 or f c DNE

Critical Number (Translation) If the function is differentiable, critical

points are those such that

Example 3Find the critical numbers of

3265)( xxxf

Example 3Find the critical numbers of

3265)( xxxf

The Closed Interval Method Idea: the absolute max/min values of a continuous function on a closed interval only occur at1. the local max/min (the critical numbers) 2. end points of the interval

The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of f at the end points.3. The largest of the values from steps 1 and 2 is the

absolute maximum value; the smallest of the those values from is the absolute minimum value.

The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of at the end points.3. The largest of the values from steps 1 and 2 is the

absolute maximum value; the smallest of the those values from is the absolute minimum value.

The Closed Interval Method To find the absolute max/min values of a continuous function on a closed interval :1. Find the values of at the critical numbers of in .2. Find the values of at the end points.3. The largest of the values from steps 1 and 2 is the

absolute maximum value; the smallest of the those values from is the absolute minimum value.

Example 4Find the absolute max/min values of

]5,3[on 112)( 3 xxxf

Expectations: Formal Conclusion