What Is the General Form of the Third Partial Derivative Test?

In summary, when discussing the second partial derivative test in multivariate calculus, there is often mention of a "higher order test" that is used when the second partial derivative test fails. It is unclear what this test is specifically referring to, but there are two possible approaches to generalize it: analyzing a bivariate cubic equation or using the determinant of a Hessian. However, both methods seem difficult and it is uncertain if they will yield the same properties as the second partial derivative test. It is also noted that as the number of variables and derivatives increases, the rank of the tensor also increases.
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rvadd
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When discussing the second partial derivative test in multivariate calculus, a reference is usually made to an elusive "higher order test" that one must defer to in the case that the second partial derivative test fails. Does anyone know the general form of these higher order test?

My first intuition would be to analyze the bivariate cubic equation (as the second derivative test is proved oftentimes via lemmas involving the bivariate quadratic equation):

[itex]ax^{3}+bx^{2}y+cxy^{2}+dy^{3}[/itex]

This approach seems difficult (involving completing a cube). In addition, I doubt the same properties will arise, since they rely on the positive values of squared expressions. At the same time, you might need to include the quadratic terms when analyzing the positive/negative/zero conditions of the concavity.

The other explanation frequently used to define the second partial derivative test is the determinant of the Hessian. This seems even worse than the previous approach because there are eight third partial derivatives and they wouldn't fit in a square matrix (which is necessary for a determinant). To generalize a Hessian, you would need to extend it into a tensor (contravariant rank 3 just as the Hessian is rank 2 and the Jacobian is rank 1 for f(x,y) ), which seems difficult once again because how then do you take the determinant (if you do need a determinant in the first place)?

Are these reasonable methods for accomplishing this generalization or am I missing some obscure/obvious simpler theorem?
 
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FAQ: What Is the General Form of the Third Partial Derivative Test?

1. What is the purpose of the Third Partial Derivative Test?

The Third Partial Derivative Test is used to determine the nature of a critical point in a function of two or more variables. It helps to classify whether the critical point is a maximum, minimum, or saddle point.

2. How is the Third Partial Derivative Test performed?

The Third Partial Derivative Test involves calculating the second-order partial derivatives of a function and evaluating them at the critical point. If the second-order derivative is positive, the critical point is a minimum. If it is negative, the critical point is a maximum. And if it is zero, further analysis is needed to determine the nature of the critical point.

3. Can the Third Partial Derivative Test be used for functions of three or more variables?

Yes, the Third Partial Derivative Test can be used for functions of any number of variables. However, it becomes more complicated as the number of variables increases.

4. What are the limitations of the Third Partial Derivative Test?

The Third Partial Derivative Test can only be applied to differentiable functions, and it can only determine the nature of a critical point. It cannot determine the existence of a critical point or the location of a global minimum or maximum.

5. How is the Third Partial Derivative Test related to the Second Partial Derivative Test?

The Third Partial Derivative Test is an extension of the Second Partial Derivative Test, which is used to determine the nature of critical points in functions of one variable. The Second Partial Derivative Test is a special case of the Third Partial Derivative Test for functions of two variables.

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