<< Chapter < Page Chapter >> Page >

The second derivative test for a function of one variable provides a method for determining whether an extremum occurs at a critical point of a function. When extending this result to a function of two variables, an issue arises related to the fact that there are, in fact, four different second-order partial derivatives, although equality of mixed partials reduces this to three. The second derivative test for a function of two variables, stated in the following theorem, uses a discriminant     D that replaces f ( x 0 ) in the second derivative test for a function of one variable.

Second derivative test

Let z = f ( x , y ) be a function of two variables for which the first- and second-order partial derivatives are continuous on some disk containing the point ( x 0 , y 0 ) . Suppose f x ( x 0 , y 0 ) = 0 and f y ( x 0 , y 0 ) = 0 . Define the quantity

D = f x x ( x 0 , y 0 ) f y y ( x 0 , y 0 ) ( f x y ( x 0 , y 0 ) ) 2 .
  1. If D > 0 and f x x ( x 0 , y 0 ) > 0 , then f has a local minimum at ( x 0 , y 0 ) .
  2. If D > 0 and f x x ( x 0 , y 0 ) < 0 , then f has a local maximum at ( x 0 , y 0 ) .
  3. If D < 0 , , then f has a saddle point at ( x 0 , y 0 ) .
  4. If D = 0 , then the test is inconclusive.

See [link] .

This figure consists of three figures labeled a, b, and c. Figure a has two bulbous mounds pointing down, and the two extrema are listed as the local minima. Figure b has two bulbous mounds pointed up, and the two extrema are listed as the local maxima. Figure c is shaped like a saddle, and in the middle of the saddle, a point is marked as the saddle point.
The second derivative test can often determine whether a function of two variables has a local minima (a), a local maxima (b), or a saddle point (c).

To apply the second derivative test, it is necessary that we first find the critical points of the function. There are several steps involved in the entire procedure, which are outlined in a problem-solving strategy.

Problem-solving strategy: using the second derivative test for functions of two variables

Let z = f ( x , y ) be a function of two variables for which the first- and second-order partial derivatives are continuous on some disk containing the point ( x 0 , y 0 ) . To apply the second derivative test to find local extrema, use the following steps:

  1. Determine the critical points ( x 0 , y 0 ) of the function f where f x ( x 0 , y 0 ) = f y ( x 0 , y 0 ) = 0 . Discard any points where at least one of the partial derivatives does not exist.
  2. Calculate the discriminant D = f x x ( x 0 , y 0 ) f y y ( x 0 , y 0 ) ( f x y ( x 0 , y 0 ) ) 2 for each critical point of f .
  3. Apply [link] to determine whether each critical point is a local maximum, local minimum, or saddle point, or whether the theorem is inconclusive.

Using the second derivative test

Find the critical points for each of the following functions, and use the second derivative test to find the local extrema:

  1. f ( x , y ) = 4 x 2 + 9 y 2 + 8 x 36 y + 24
  2. g ( x , y ) = 1 3 x 3 + y 2 + 2 x y 6 x 3 y + 4
  1. Step 1 of the problem-solving strategy involves finding the critical points of f . To do this, we first calculate f x ( x , y ) and f y ( x , y ) , then set each of them equal to zero:
    f x ( x , y ) = 8 x + 8 f y ( x , y ) = 18 y 36 .

    Setting them equal to zero yields the system of equations
    8 x + 8 = 0 18 y 36 = 0 .

    The solution to this system is x = −1 and y = 2 . Therefore ( −1 , 2 ) is a critical point of f .
    Step 2 of the problem-solving strategy involves calculating D . To do this, we first calculate the second partial derivatives of f :
    f x x ( x , y ) = 8 f x y ( x , y ) = 0 f y y ( x , y ) = 18 .

    Therefore, D = f x x ( −1 , 2 ) f y y ( −1 , 2 ) ( f x y ( −1 , 2 ) ) 2 = ( 8 ) ( 18 ) ( 0 ) 2 = 144 .
    Step 3 states to check [link] . Since D > 0 and f x x ( −1 , 2 ) > 0 , this corresponds to case 1. Therefore, f has a local minimum at ( −1 , 2 ) as shown in the following figure.
    The function f(x, y) = 4x2 + 9y2 + 8x – 36y + 24 is shown with local minimum at (–1, 2, –16). The shape is a plane curving up on both ends parallel to the y axis.
    The function f ( x , y ) has a local minimum at ( −1 , 2 , −16 ) .
  2. For step 1, we first calculate g x ( x , y ) and g y ( x , y ) , then set each of them equal to zero:
    g x ( x , y ) = x 2 + 2 y 6 g y ( x , y ) = 2 y + 2 x 3 .

    Setting them equal to zero yields the system of equations
    x 2 + 2 y 6 = 0 2 y + 2 x 3 = 0 .

    To solve this system, first solve the second equation for y. This gives y = 3 2 x 2 . Substituting this into the first equation gives
    x 2 + 3 2 x 6 = 0 x 2 2 x 3 = 0 ( x 3 ) ( x + 1 ) = 0 .

    Therefore, x = −1 or x = 3 . Substituting these values into the equation y = 3 2 x 2 yields the critical points ( −1 , 5 2 ) and ( 3 , 3 2 ) .

    Step 2 involves calculating the second partial derivatives of g :

    g x x ( x , y ) = 2 x g x y ( x , y ) = 2 g y y ( x , y ) = 2 .

    Then, we find a general formula for D :
    D = g x x ( x 0 , y 0 ) g y y ( x 0 , y 0 ) ( g x y ( x 0 , y 0 ) ) 2 = ( 2 x 0 ) ( 2 ) 2 2 = 4 x 0 4 .

    Next, we substitute each critical point into this formula:
    D ( −1 , 5 2 ) = ( 2 ( −1 ) ) ( 2 ) ( 2 ) 2 = −4 4 = −8 D ( 3 , 3 2 ) = ( 2 ( 3 ) ) ( 2 ) ( 2 ) 2 = 12 4 = 8 .

    In step 3, we note that, applying [link] to point ( −1 , 5 2 ) leads to case 3 , which means that ( −1 , 5 2 ) is a saddle point. Applying the theorem to point ( 3 , 3 2 ) leads to case 1, which means that ( 3 , 3 2 ) corresponds to a local minimum as shown in the following figure.
    The function f(x, y) = (1/3)x3 + y2 + + 2xy – 6x – 3y + 4 is shown with local minimum at (3, –3/2, –29/4) and saddle point at (−1, 5/2, 41/12). The shape is a plane curving up on the corners near (4, 3) and (−2, −2).
    The function g ( x , y ) has a local minimum and a saddle point.
Got questions? Get instant answers now!

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com 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?

Ask