<< Chapter < Page Chapter >> Page >

The proof of the extreme value theorem is beyond the scope of this text. Typically, it is proved in a course on real analysis. There are a couple of key points to note about the statement of this theorem. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. If the interval I is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over I . For example, consider the functions shown in [link] (d), (e), and (f). All three of these functions are defined over bounded intervals. However, the function in graph (e) is the only one that has both an absolute maximum and an absolute minimum over its domain. The extreme value theorem cannot be applied to the functions in graphs (d) and (f) because neither of these functions is continuous over a closed, bounded interval. Although the function in graph (d) is defined over the closed interval [ 0 , 4 ] , the function is discontinuous at x = 2 . The function has an absolute maximum over [ 0 , 4 ] but does not have an absolute minimum. The function in graph (f) is continuous over the half-open interval [ 0 , 2 ) , but is not defined at x = 2 , and therefore is not continuous over a closed, bounded interval. The function has an absolute minimum over [ 0 , 2 ) , but does not have an absolute maximum over [ 0 , 2 ) . These two graphs illustrate why a function over a bounded interval may fail to have an absolute maximum and/or absolute minimum.

Before looking at how to find absolute extrema, let’s examine the related concept of local extrema. This idea is useful in determining where absolute extrema occur.

Local extrema and critical points

Consider the function f shown in [link] . The graph can be described as two mountains with a valley in the middle. The absolute maximum value of the function occurs at the higher peak, at x = 2 . However, x = 0 is also a point of interest. Although f ( 0 ) is not the largest value of f , the value f ( 0 ) is larger than f ( x ) for all x near 0. We say f has a local maximum at x = 0 . Similarly, the function f does not have an absolute minimum, but it does have a local minimum at x = 1 because f ( 1 ) is less than f ( x ) for x near 1.

The function f(x) is shown, which curves upward from quadrant III, slows down in quadrant II, achieves a local maximum on the y-axis, decreases to achieve a local minimum in quadrant I at x = 1, increases to a local maximum at x = 2 that is greater than the other local maximum, and then decreases rapidly through quadrant IV.
This function f has two local maxima and one local minimum. The local maximum at x = 2 is also the absolute maximum.


A function f has a local maximum    at c if there exists an open interval I containing c such that I is contained in the domain of f and f ( c ) f ( x ) for all x I . A function f has a local minimum    at c if there exists an open interval I containing c such that I is contained in the domain of f and f ( c ) f ( x ) for all x I . A function f has a local extremum    at c if f has a local maximum at c or f has a local minimum at c .

Note that if f has an absolute extremum at c and f is defined over an interval containing c , then f ( c ) is also considered a local extremum. If an absolute extremum for a function f occurs at an endpoint, we do not consider that to be a local extremum, but instead refer to that as an endpoint extremum.

Given the graph of a function f , it is sometimes easy to see where a local maximum or local minimum occurs. However, it is not always easy to see, since the interesting features on the graph of a function may not be visible because they occur at a very small scale. Also, we may not have a graph of the function. In these cases, how can we use a formula for a function to determine where these extrema occur?

Questions & Answers

I don't understand the formula
Adaeze Reply
who's formula
What is a independent variable
Sifiso Reply
a variable that does not depend on another.
solve number one step by step
bil Reply
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
1-sin x/cos x= cos x/-1+sin x
how to prove 1-sun x/cos x= cos x / -1+sin x?
how to prove tan^2 x=csc^2 x tan^2 x-1?
Rochel Reply
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
how to prove sin x - sin x cos^2 x=sin^3x?
Rochel Reply
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
how to prove tan^2 x/tan^2 x+1= sin^2 x
not a bad question
what is function.
Nawaz Reply
what is polynomial
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
what is the power rule
Vanessa Reply
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
The set of inputs of a function. x goes in the function, y comes out.
where u from verna
If you differentiate then answer is not x
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
what is functions
mahin Reply
give different types of functions.
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
differentiate each term
Practice Key Terms 9

Get the best Calculus volume 1 course in your pocket!

Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?