Hi

I have two functions that I need to find the maximum and minimum for, I have read about it the methods, but I find it hard to understand.

I have two functions that I need to find the critical points of:





all I know is that you are able to solve these functions

can someone briefly explain to me how to find the critical points and how to tell if they are a minimum, maximum or saddlepoint?

im not really a pro at math but i think the basic theory is that the deviation of a function is the function that gives you the amount of how much the function really is rising (or negatively rising for that matter)
so if the deviation of your function f' = 0 then you have either a minimum, a maximum or a saddle point
to get to know what it is you can look at the limes of those places, seeing if the deviation is negative before the point and positive after after ( meaning you have a minimum, if you look at the deviation f' (not the function)) or the other way round (meaning you have a maximum) or if they stay the same, meaning you have a saddlepoint

however i am absolutely unsure how that converts to functions with multiple variables.

differentiate with respect to one variable while leaving the other constant

differentiate with respect to the other variable while leaving the first constant

then youll get like a system of equations and youll have to solve it for the critical values

then test the values

Critical points are where the partial derivatives are equal to zero. Graphically it would be where the function levels off at a max, min or saddle point. A saddle point would be like a point of inflection in a curve.

Once you determine the critical points, you need to test each with a formula to determine whether it's a max, min or saddle point. It's a little difficult to write in a text box (you should have it in your book).

(Fxx)(Fyy) - (Fxy)^2 = D (those are 2nd order partial derivatives)

If D > 0 and Fxx > 0, min
If D > 0 and Fxx < 0, max
If D < 0, saddle point
If D = 0, inconclusive

Thank you for your help