MTH 251 – Differential Calculus

Chapter 4 – Applications of Derivatives

Section 4.1

Extreme Values of Functions

Copyright © 2010 by Ron Wallace, all rights reserved.

Terminology

• The following are the same … Absolute (aka: Global) Extrema Absolute (aka: Global) Minimum and/or Maximum Extreme Values

• Basic problem of this chapter … Determine the extreme values of a function over an

interval. i.e. Given f(x) where x [a,b] or (a,b) or [a,b) or (a,b];

determine the largest and/or smallest value of f(x).• Note: The extreme values are values of the function.

The extreme values occur at one or more values of x in the interval.

Max & Min – Formal Definitions• f(x) has an Absolute Maximum over a domain D at

a point x = c if and only if f(x) ≤ f(c) for all x in D.

• f(x) has an Absolute Minimum over a domain D at a point x = c if and only if f(x) ≥ f(c) for all x in D.

Note: Absolute Extrema may occur at more than one value of x.

Possible Locations of Extrema• Top of a peak

• Bottom of a valley

• End point

• Point of discontinuity the function must be defined

Do Extrema Exist?

• Possibilities … Both max & min? Max but no min? Min but no max? No max or min?

• The Extreme Value Theorem If f(x) is continuous over (aka: on) [a,b], then f(x)

has a absolute maximum value M and an absolute minimum value m over the interval.

• Note that m ≤ f(x) ≤ M for all x [a,b] and … • … there exists x1 & x2 [a,b] where f(x1) = m and

f(x2) = M

Local Extrema

• If there is some open interval that contains x = c where f(c) is an extrema over that interval, then f(c) is a Local Extrema. aka: Relative Extrema

• The left endpoint of the domain of a function is a local extrema.

• A right endpoint of the domain of a function is a local extrema.

Finding Extrema• Some facts …

Absolute extrema are also relative extrema. Possible locations of relative extrema are the same

as absolute extrema• i.e. peaks, valleys, endpoints, discontinuities

Peaks & Valleys occur at “critical points”• Points where f ’(x) is zero or undefined• Note: Not all critical points are extrema

Proof regarding Critical Points• If f(c) is a local maximum and f’(c) exists, then

f’(c) = 0. Local Max implies that f(x) ≤ f(c) for some interval

containing c. That is, f(x) – f(c) ≤ 0

( ) ( )'( ) limx c

f x f cf cx c

( ) ( )'( ) limx c

f x f cf cx c

0

0

Since these must be equal …

'( ) 0f c

The proof for local minimums would be essentially the same (all of the inequalities would be reversed).

Finding Extrema• Some facts …

Absolute extrema are also relative extrema. Possible locations of relative extrema are the same

as absolute extrema• i.e. peaks, valleys, endpoints, discontinuities

Peaks & Valleys occur at “critical points”• Points where f ’(x) is zero or undefined• Note: Not all critical points are extrema

• Method … for closed intervals1. Find the values of x of all critical points.

• i.e. f ’(x) = 0 or DNE§ Calculate f(x) for all critical points and endpoints.§ The extrema are the largest and smallest of the

values in step 2.

Finding Extrema – Example• Method … for closed intervals

1. Find the values of x of all critical points.• i.e. f ’(x) = 0 or DNE

2. Calculate f(x) for all critical points and endpoints.3. The extrema are the largest and smallest of the

values in step 2.

• Determine the extrema for …3 2( ) 2 9 , [ 1,5]f x x x x

20

3

Extrema on Open Intervals

• Instead of calculating the value of the function at the endpoint, you must calculate the limit as x approaches the endpoint.

• Method … for open intervals1. Find the values of x of all critical points.

• i.e. f ’(x) = 0 or DNE2. Calculate f(x) for all critical points.3. Calculate the limits at the endpoints.

• one sided limits4. The extrema are the largest and smallest of the

values in step 2 provided that they are larger or smaller than the limits in step 3.

Note: Semi-open intervals will use a combination of the two previous cases.

Finding Extrema – Example

2

1( )1

f xx

Domain?

• Method … for open intervals1. Find the values of x of all critical points.

• i.e. f ’(x) = 0 or DNE2. Calculate f(x) for all critical points.3. Calculate the limits at the endpoints.

• one sided limits4. The extrema are the largest and smallest of the

values in step 2 provided that they are larger or smaller than the limits in step 3.

• Determine the extrema for …

• Determine the extrema for … (note: semi-open)

Finding Extrema – Example

2( ) ( 1) , [ 2,2)f x x x

• Determine the extrema for … (note: open … domain?)

Finding Extrema – Example

3 2( ) 3f x x x