<< Chapter < Page Chapter >> Page >

Use the second derivative to find the local extrema of the function

f ( x , y ) = x 3 + 2 x y 6 x 4 y 2 .

( 4 3 , 1 3 ) is a saddle point, ( 3 2 , 3 8 ) is a local maximum.

Got questions? Get instant answers now!

Absolute maxima and minima

When finding global extrema of functions of one variable on a closed interval, we start by checking the critical values over that interval and then evaluate the function at the endpoints of the interval. When working with a function of two variables, the closed interval is replaced by a closed, bounded set. A set is bounded if all the points in that set can be contained within a ball (or disk) of finite radius. First, we need to find the critical points inside the set and calculate the corresponding critical values. Then, it is necessary to find the maximum and minimum value of the function on the boundary of the set. When we have all these values, the largest function value corresponds to the global maximum and the smallest function value corresponds to the absolute minimum. First, however, we need to be assured that such values exist. The following theorem does this.

Extreme value theorem

A continuous function f ( x , y ) on a closed and bounded set D in the plane attains an absolute maximum value at some point of D and an absolute minimum value at some point of D .

Now that we know any continuous function f defined on a closed, bounded set attains its extreme values, we need to know how to find them.

Finding extreme values of a function of two variables

Assume z = f ( x , y ) is a differentiable function of two variables defined on a closed, bounded set D . Then f will attain the absolute maximum value and the absolute minimum value, which are, respectively, the largest and smallest values found among the following:

  1. The values of f at the critical points of f in D .
  2. The values of f on the boundary of D .

The proof of this theorem is a direct consequence of the extreme value theorem and Fermat’s theorem. In particular, if either extremum is not located on the boundary of D , then it is located at an interior point of D . But an interior point ( x 0 , y 0 ) of D that’s an absolute extremum is also a local extremum; hence, ( x 0 , y 0 ) is a critical point of f by Fermat’s theorem. Therefore the only possible values for the global extrema of f on D are the extreme values of f on the interior or boundary of D .

Problem-solving strategy: finding absolute maximum and minimum values

Let z = f ( x , y ) be a continuous function of two variables defined on a closed, bounded set D , and assume f is differentiable on D . To find the absolute maximum and minimum values of f on D , do the following:

  1. Determine the critical points of f in D .
  2. Calculate f at each of these critical points.
  3. Determine the maximum and minimum values of f on the boundary of its domain.
  4. The maximum and minimum values of f will occur at one of the values obtained in steps 2 and 3 .

Finding the maximum and minimum values of f on the boundary of D can be challenging. If the boundary is a rectangle or set of straight lines, then it is possible to parameterize the line segments and determine the maxima on each of these segments, as seen in [link] . The same approach can be used for other shapes such as circles and ellipses.

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?