How Do Lagrange Multipliers Relate to Critical Points of a Function?

[bitrex]

I have a problem where I'd like to minimize a certain function subject to the constraint that a related function is at a maximum, that is I have a function F(a,b) I would like to know what its minimum is when G(a,b) is at a maximum. I'm not sure how to set this problem up, I know that for the function G to have a maximum I have to apply the second partial derivative test but I am not sure how to translate this test into a form useful as a constraint for a Lagrange multiplier problem. Thanks for any advice!

 

[HallsofIvy]

What, precisely, is G(a,b)? You say you want to minimize F(a, b) when G(a,b) is a maximum but if G(a,b) is maximum at a specific point (a, b), the requires F(a,b) to be a specific number and cannot be "minimized". This problem only makes sense when G(a,b) reaches a maximum at every point of some curve or other set.

 

[bitrex]

Thank you for your reply, and sorry for the delay in mine. What I really wanted to do in my problem was find the maximum value of the difference between two functions, and to do that Lagrange multipliers weren't really necessary. I guess I just had a hammer looking for a nail.

My mistake does lead me to a related question about Lagrange multipliers though - if one takes a function A of say two variables and sets their partial derivatives equal to zero, and finds that these critical points are minima, maxima, or a saddle point of the function, and then uses those partial derivatives equal to zero as the constraint functions B and C for another function D in a Lagrange multiplier problem - what geometrical interpretation does the minimized function with those constraints have to the original function A, if any?