MAT 1234Calculus I

Section 3.1

Maximum and Minimum Values

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WebAssign 3.1 Quiz– 2.7, 2.9

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)

f has an absolute maximum at x=c on D if

for all x in D (D =Domain of f) )()( xfcf

c

D

Definition (Absolute Min)

f has an absolute minimum at x=c on D if

for all x in D (D =Domain of f) )()( xfcf

c

D

Definition

The absolute maximum and minimum values of f are called the extreme values of f.

x

Example 1y

Absolute max.

Absolute min.

Definition (Local Max/Min)

f has an local maximum at x=c if

for all x in some open interval containing c

)()( xfcf

f has an local minimum at x=c if

for all x in some open interval containing c

)()( xfcf

x

Example 1y

Local max.

Local min.

Q&A

An end point is not a local max/min, why?

The Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains an absolute max value f(c) and an absolute min value f(d) at some numbers c and d in [a,b].

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

The Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains an absolute max value f(c) and an absolute min value f(d) at some numbers c and d in [a,b].

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)

f is not continuous on [a,b]

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 f is continuous on a closed interval [a,b].

Next: How to find them?

Fermat’s Theorem

If f has a local maximum or minimum at c, and if exists, then)(cf 0)( cf

c x

y

Q&A: T or F

The converse of the theorem: If , then f has a local maximum or minimum at c

0)( cf

Definition (Critical Number)

A critical number of a function f is a number c in the domain of f such that either or does not exist.0)( cf )(cf

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 c such that ( ) 0f c

Example 3

Find the critical numbers of 3265)( xxxf

Example 3

Find the critical numbers of 3265)( xxxf

The Closed Interval Method

Idea: the absolute max/min values of a continuous function f on a closed interval [a,b] only occur at

1. 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 f on a closed interval [a,b]:

1. Find the values of f at the critical numbers of f in (a,b).

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 f on a closed interval [a,b]:

1. Find the values of f at the critical numbers of f in (a,b).

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 f on a closed interval [a,b]:

1. Find the values of f at the critical numbers of f in (a,b).

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.

Example 4

Find the absolute max/min values of

]5,3[on 112)( 3 xxxf

Expectations: Formal Conclusion

Classwork

Do part (a) only.