Question

write the rules for second order partial derivative test for local extreme values.

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

The second partial derivative test is a method in multi-variable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.

Functions of two variables

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

$\bold{{\displaystyle H(x,y)={\begin{pmatrix}f_{xx}(x,y)&f_{xy}(x,y)\\f_{yx}(x,y)&f_{yy}(x,y)\end{pmatrix}}}}$

Define D(x, y) to be the determinant

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

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

• If ${\displaystyle D(a,b)>0}$ and ${\displaystyle f_{xx}(a,b)>0}$ then ${\displaystyle (a,b)}$ is a local minimum of f.
• If ${\displaystyle D(a,b)>0}$ and ${\displaystyle f_{xx}(a,b)<0}$ then ${\displaystyle (a,b)}$ is a local maximum of f.
• If ${\displaystyle D(a,b)<0}$ then ${\displaystyle (a,b)}$ is a saddle point of f.
• If ${\displaystyle D(a,b)=0}$ then the second derivative test is inconclusive, and the point ${\displaystyle (a,b)}$ 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 fxx + fyy in place of the value fxx in the first two cases above.[citation needed] Such variations in the procedure applied do not alter the outcome of the test.

Functions of many variables

For a function f of three or more 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 for which the Hessian matrix is invertible:

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

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

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 that in the one-variable case, the Hessian condition simply gives the usual second derivative test.

In the two variable case, ${\displaystyle D(a,b)}$ and${\displaystyle f_{xx}(a,b)}$ 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 (Sylvester's criterion): 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.