Maximum and minimum values section 3 1
This presentation is the property of its rightful owner.
Sponsored Links
1 / 24

Maximum and Minimum Values ( Section 3.1) PowerPoint PPT Presentation


  • 110 Views
  • Uploaded on
  • Presentation posted in: General

Maximum and Minimum Values ( Section 3.1). Alex Karassev. Absolute maximum values. A function f has an absolute maximum value on a set S at a point c in S if f(c) ≥ f(x) for all x in S. y. y = f(x). f(c). x. S. c. Absolute minimum values.

Download Presentation

Maximum and Minimum Values ( Section 3.1)

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Maximum and minimum values section 3 1

Maximum and Minimum Values(Section 3.1)

Alex Karassev


Absolute maximum values

Absolute maximum values

  • A function f has an absolute maximum value on a set S at a point c in S if f(c) ≥ f(x) for all x in S

y

y = f(x)

f(c)

x

S

c


Absolute minimum values

Absolute minimum values

  • A function f has an absolute minimum value on a set S at a point c in S if f(c) ≤ f(x) for all x in S

y

y = f(x)

x

f(c)

S

c


Example f x x 2

Example: f(x) = x2

  • S = (-∞, ∞)

  • No absolute maximum

  • Absolute minimum:f(0) = 0 at c = 0

y

x

0


Example f x x 21

Example: f(x) = x2

  • S = [0,1]

  • Absolute maximumf(1) = 1 at c = 1

  • Absolute minimum:f(0) = 0 at c = 0

y

x

0

1


Example f x x 22

Example: f(x) = x2

  • S = (0,1]

  • Absolute maximumf(1) = 1 at c = 1

  • No absoluteminimum,although function isbounded from below:0 < x2 for allx in (0,1] !

y

x

0

1


Local maximum values

Local maximum values

  • A function f has a local maximum value at a point c if f(c) ≥ f(x) for all x near c (i.e. for all x in some open interval containing c)

y

y = f(x)

x

c


Local minimum values

Local minimum values

  • A function f has a local minimum value at a point cif f(c) ≤ f(x) for all x near c(i.e. for all x in some open interval containing c)

y

y = f(x)

x

c


Example y sin x

Example: y = sin x

f(x) = sin xhas local (and absolute) maximumat all points of the form π/2 + 2πk,and local (and absolute) minimumat all points of the form -π/2 + 2πk,where k is an integer

1

- π/2

π/2

-1


Applications

Applications

  • Curve sketching

  • Optimization problems (with constraints),for example:

    • Finding parameters to minimizemanufacturing costs

    • Investing to maximize profit (constraint: amount of money to invest is limited)

    • Finding route to minimize the distance

    • Finding dimensions of containers to maximize volumes (constraint: amount of material to be used is limited)


Extreme value theorem

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attainsabsolute maximum value f(cMAX) andabsolute minimum value f(cMIN)at some numbers cMAX andcMIN in [a,b]


Extreme value theorem examples

Extreme Value Theorem - Examples

y

y

y = f(x)

y = f(x)

x

x

a

a

b

cMIN

cMAX

cMIN

cMAX= b

Both absolute max and absolute min are attained in the open interval (a,b) at the points of local max and min

Absolute maximum is attained at the right end point: cMAX = b


Continuity is important

Continuity is important

y

x

-1

1

0

No absolute maximum or minimumon [-1,1]


Closed interval is important

Closed interval is important

  • f(x) = x2, S = (0,1]

  • No absoluteminimum in (0,1]

y

x

0

1


How to find max and min values

How to find max and min values?

  • Absolute maximum or minimum values of a function, continuous on a closed interval are attained either at the points which are simultaneously the points of local maximum or minimum, or at the endpoints

  • Thus, we need to know how to find points of local maximums and minimums


Fermat s theorem

Fermat's Theorem

  • If f has a local maximum or minimum at c and f′(c) exists, then f′(c) = 0

y

horizontal tangent line at the point of local max (or min)

y = f(x)

x

c


Converse of fermat s theorem does not hold

Converse of Fermat's theoremdoes not hold!

  • If f ′(c) = 0 it does not mean that c is a point of local maximum or minimum

  • Example: f(x) = x3, f ′(0) = 0, but 0 is not a point of local max or min

  • Nevertheless, points c wheref ′(c) = 0 are "suspicious" points(for local max or min)

y

x


Problem f not always exists

Problem: f′ not always exists

  • f(x) = |x|

  • It has local (and absolute) minimum at 0

  • However, f′ (0) does not exists!

y

x


Critical numbers

Critical numbers

  • Two kinds of "suspicious" points(for local max or min):

    • f′(c) = 0

    • f′(c) does not exists


Critical numbers definition

Critical numbers – definition

  • A number c is called a critical number of function f if the following conditions are satisfied:

    • c is in the domain of f

    • f′(c) = 0 or f′(c) does not exist


Closed interval method

Closed Interval Method

  • The method to find absolute maximum or minimum of a continuous function, defined on a closed interval [a,b]

  • Based on the fact that absolute maximum or minimum

    • either is attained at some point inside the open interval (a,b) (then this point is also a point of local maximum or minimum and hence isa critical number)

    • or is attained at one of the endpoints


Closed interval method1

Closed Interval Method

  • To find absolute maximum and minimumof a function f, continuous on [a,b]:

    • Find critical numbers inside (a,b)

      • Find derivative f′ (x)

      • Solve equation f′ (x)=0 for x and choose solutions which are inside (a,b)

      • Find numbers in (a,b) where f′ (x) d.n.e.

    • Suppose that c1, c2, …, ckare all critical numbers in (a,b)

      • The largest of f(a), f(c1), f(c2), …, f(ck), f(b) is theabsolute maximum of f on [a,b]

      • The smallest of these numbers is theabsolute minimum of f on [a,b]


Example

Example

  • Find the absolute maximum and minimum values of f(x) = x/(x2+1) on the interval [0,2]


Solution

Find the absolute maximum and minimum values of f(x) = x/(x2+1) on the interval [0,2]

Solution

  • Find f′(x):

  • Critical numbers: f′(x) = 0 ⇔ 1– x2 = 0

  • So x = 1 or x = – 1

  • However, only 1 is inside [0,2]

  • Now we need to compare f(0), f(1), and f(2):

  • f(0) = 0, f(1) = 1/2, f(2)= 2/5

  • Therefore 0 is absolute minimum and 1/2 is absolute maximum


  • Login