calc 3 question - lagrange multiplier - what is max or min?

A feasible point x* is a local constrained minimum if for all feasible changes about x* the change in the objective function is always positive. If x* is also a solution of the Lagrangian equations then this can be shown to be equivalent to the requirement that the Lagrangian function only increases along the linearised constraints at x*. In other words the projected Lagrangian has only positive curvature at x*. Similar statements apply for a constrained maximum.

Taken literally, this can be daunting to confirm but some practical approaches are

(a) If x* is unique then compare f(x*) with the value of f at another feasible point.

(b) If x* are multiple compare the value of f at some nearby feasible points. This will give indications as to whether each x* is a maximum or minimum. With one degree of freedom, say two variables and one constraint, and a perturbation “either side” of x*, you can be sure of the result.

(c) Examine the “Bordered Hessian” for the problem. This is constructed from the second-derivative matrix ( Hessian) of the Lagrangian, bordered with constraint gradient information. By testing the values of a series of related determinants you can check for a minimum or maximum. This test is equivalent to demonstrating that the projected Hessian of the Lagrangian is positive/negative definite. There's an easy-to-understand statement of the method with examples here ...

http://www.tbparis.com/Econ130/Chiang%20-%20Chapte...

and a more detailed exposition here ...

http://www2.econ.iastate.edu/classes/econ500/halla...

In practice numerical examples are quite straightforward.

Without constraints and only two variables this method reduces to the second partial derivative test used in uncontrained optimisation. See ...

http://en.wikipedia.org/wiki/Second_partial_deriva...

take the sq. root to hit upon y=+/-sqrt(9+xz) look widely used? it quite is a sphere. plug in 0's for x and z and that'll get you a circle with radius 3 parallel to the y plane. As for the Lagrange multipliers, look on your e book for the examples. also effectual/you in all likelihood already comprehend say you want dy/dz, pretend that each and each element that ought to not a z is a continuing