Binomial transform

In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.

The test

Functions of two variables

Suppose that f(x, y) is a differentiable real function of two variables whose second partial derivatives exist. The Hessian matrix H of f is the 2 × 2 matrix of partial derivatives of f:

$H(x,y)=\left(\begin{array}{cc}{f}_{xx}(x,y)& f_{xy}(x,y)\\ f_{yx}(x,y)& f_{yy}(x,y)\end{array}\right)$.

Define D(x, y) to be the determinant

$D(x,y)=\det (H(x,y))=f_{xx}(x,y)f_{yy}(x,y)-{(f_{xy}(x,y))}^2$,

of H. Finally, suppose that (a, b) is a critical point of f (that is, f_x(a, b) = f_y(a, b) = 0). Then the second partial derivative test asserts the following:

1. If $D(a,b)>0$ and $f_{xx}(a,b)>0$ then $(a,b)$ is a local minimum of f.
2. If $D(a,b)>0$ and $f_{xx}(a,b)<0$ then $(a,b)$ is a local maximum of f.
3. If $D(a,b)<0$ then $(a,b)$ is a saddle point of f.
4. If $D(a,b)=0$ then the second derivative test is inconclusive, and the point (a, b) could be any of a minimum, maximum or saddle point.

Note that other equivalent versions of the test are possible. For example, some texts may use the trace f_xx + f_yy in place of the value f_xx in the first two cases above.Template:Cn Such variations in the procedure applied do not alter the outcome of the test.

Functions of many variables

For a function f of more than two variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point (a, b, ...) for which the Hessian matrix is invertible:

1. If the Hessian is positive definite (equivalently, has all eigenvalues positive) at (a, b, ...), then f attains a local minimum at (a, b, ...).
2. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at (a, b, ...), then f attains a local maximum at (a, b, ...).
3. If the Hessian has both positive and negative eigenvalues then (a, b, ...) is a saddle point for f (and in fact this is true even if (a, b, ...) is degenerate).

In those cases not listed above, the test is inconclusive.

Note that for functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point, because the number of jointly sufficient second-order conditions is equal to the number of variables, and the sign condition on the determinant of the Hessian is only one of the conditions. Note also that this statement of the second derivative test for many variables also applies in the two-variable and one-variable case. In the latter case, we recover the usual second derivative test.

In the two variable case, $D(a,b)$ and $f_{xx}(a,b)$ are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian. For the general case of an arbitrary number n of variables, there are n sign conditions on the n principal minors of the Hessian matrix that together are equivalent to positive or negative definiteness of the Hessian: for a local minimum, all the principal minors need to be positive, while for a local maximum, the minors with an odd number of rows and columns need to be negative and the minors with an even number of rows and columns need to be positive. See Hessian matrix#Bordered Hessian for a discussion that generalizes these rules to the case of equality-constrained optimization.

Geometric interpretation in the two-variable case

Before you choose any particular company it is vital to understand in full how the different plans can vary. There is no other better method than to create a message board so that people can relax and "chill" on your website and check out your articles more. You should read the HostGator review, even before registering with a web hosting company. but Hostgator in addition considers the surroundings. You can even use a Hostgator reseller coupon for unlimited web hosting at HostGator! Most of individuals by no means go for yearly subscription and choose month to month subscription. Several users commented that this was the deciding factor in picking HostGator but in any case there is a 45 day Money Back Guarantee and there is no contract so you can cancel at any time. GatorBill is able to send you an email notice about the new invoice. In certain cases a dedicated server can offer less overhead and a bigger revenue in investments. With the plan come a Free Billing Executive, Free sellers account and Free Hosting Templates.



This is one of the only things that require you to spend a little money to make money. Just go make an account, get a paypal account, and start selling. To go one step beyond just affiliating products and services is to create your own and sell it through your blog. Not great if you really enjoy trying out all the themes. Talking in real time having a real person causes it to be personal helping me personally to sort out how to proceed. The first step I took was search for a discount code, as I did with HostGator. Using a HostGator coupon is a beneficial method to get started. As long as the necessities are able to preserve the horizontal functionality of your site, you would pretty much be fine. Assume that all derivatives of f are evaluated at (a, b), and that the values of the first derivatives vanish there.

If $D<0$ then $f_{xx}{f}_{yy}<f_{xy}^2$. If $f_{xx}$ and $f_{yy}$ have different signs, then one must be positive and the other must be negative. Thus the concavities of the x cross section (the yz trace) and the y cross section (the xz trace) are in opposite direction. This is clearly a saddle point.

If $D>0$ then $f_{xx}{f}_{yy}>f_{xy}^2$, which implies that $f_{xx}$ and $f_{yy}$ are the same sign and sufficiently large. For this case the concavities of the x and y cross sections are either both up if positive, or both down if negative. This is clearly a local minimum or a local maximum, respectively.

This leaves the last case of D < 0 — so $f_{xx}{f}_{yy}<f_{xy}^2$ — and $f_{xx}$ and $f_{yy}$ having the same sign. The geometric interpretation of what is happening here is that since $f_{xy}$ is large it means the slope of the graph in one direction is changing rapidly as we move in the orthogonal direction and overcoming the concavity of the orthogonal direction. So for example, let's take the case of all second derivatives are positive and (a,b) = (0,0). In the case of D > 0 it would mean that any direction in the xy plane we move from the origin, the value of the function increases--a local minimum. In the D < 0 case ($f_{xy}$ sufficiently large), however, if we move at some direction between the x and y axis into the second quadrant, for example, of the xy plane, then despite the fact that the positive concavity would cause us to expect the value of the function to increase, the slope in the x direction is increasing even faster, which means that as we go left (negative x-direction) into the second quadrant, the value of the function ends up decreasing. Additionally, since the origin is a stationary point by hypothesis, we have a saddle point.

Examples

critical points of $f(x,y)=(x+y)(xy+x{y}^2)$
maxima (red) and saddle points (blue)

To find and classify the critical points of the function

$z=f(x,y)=(x+y)(xy+x{y}^2)$,

we first set the partial derivatives

$\frac{\partial z}{\partial x}=y(2x+y)(y+1)$ and $\frac{\partial z}{\partial y}=x(3y^2+2y(x+1)+x)$

equal to zero and solve the resulting equations simultaneously to find the four critical points

$(0,0),(0,-1),(1,-1)$ and $(\frac{3}{8},-\frac{3}{4})$.

In order to classify the critical points, we examine the value of the determinant D(x, y) of the Hessian of f at each of the four critical points. We have

$\begin{array}{rl}D(a,b)& =f_{xx}(a,b)f_{yy}(a,b)-{(f_{xy}(a,b))}^2\\ & =2b(b+1)\cdot 2a(a+3b+1)-(2a+2b+4ab+3b^2{)}^2.\end{array}$

Now we plug in all the different critical values we found to label them; we have

$D(0,0)=0;D(0,-1)=-1;D(1,-1)=-1;D(\frac{3}{8},-\frac{3}{4})=\frac{27}{128}.$

Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at $(\frac{3}{8},-\frac{3}{4})$ since $f_{xx}=-\frac{3}{8}<0$. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Relative Minimums and Maximums - Paul's Online Math Notes - Calc III Notes (Lamar University)
• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.