Constrained Extrema and Lagrange Multipliers

[throneoo]

Suppose I have a function f(x,y) I would like to optimize, subject to constraint g(x,y)=0.

Let H=f+λg,

The extrema occurs at (x,y) which satisfy
H_y=0
H_x=0
g(x,y)=0

Suppose the solutions are (a,b) and (c,d).

If f(a,b)=f(c,d) , how do I determine whether they are maxima or minima?

 

[mathman]

One way is brute force. Evaluate the function slightly away from the test point.