# Maxima and Minima of a Function of Several Variables.

#### beezee99

Hello all.
Please have a look at the function below. Here A,B and C are constants while x, y and z are independent variables. Please note that this is a general representation and the variables do not refer to space and time. I want to find the critical points for determination of maxima and minima of this function. Can anyone help me on this one? Thanks.

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#### Ackbeet

MHF Hall of Honor
Are you trying to find the extrema over some region of R^3?

#### HallsofIvy

MHF Helper
Take the partial derivatives and set them equal to 0. Solve those equations for x, y, and z. Can you find the partial derivatives?

#### mfetch22

Hello all.
Please have a look at the function below. Here A,B and C are constants while x, y and z are independent variables. Please note that this is a general representation and the variables do not refer to space and time. I want to find the critical points for determination of maxima and minima of this function. Can anyone help me on this one? Thanks.

Okay, to find the critical points of a function of multiple variables you have to find the zeros of the partial derivitves like the poster above me said. To take a partial derivitive with respect to a specific variable you treat all the other variables as if they were constants. So if we had:

The partial derivite with respect to x is written as
and is taken by treating y as a constant. Since y^2 is simply a "constant" squared, it is also a constant, and will be treated as such. This is what the partial derivitive comes out to:

Simmilarly with respect to y we have:

For my exapmle, to find the critical points I simply find the zeros of these functions. This apllies to your 3 varible equation also. Hope this helps.

#### beezee99

Thanks.

Hi.
Thank you mfetch22 for the example.

For Ackbeet's question, my variable x lie in the range of 0.4 and 1 while f lies in the range of 0.5 and 1.

In my case, if I take derivatives w.r.t. x, y and z. Then I set them equal to zero. So can I get any value of any of the variable by comparison of any two derivative. Say if I compare derivatives w.r.t. x and z with each other, and I get the value of z in terms of x and y. So does that value apply to the original function as well in the ranges specified.

Thanks.

#### Ackbeet

MHF Hall of Honor
Usually, for a problem like this, there are intervals for all three variables x, y, and z. It's a bit unusual to have an interval for f, although I suppose you could. That might impose a constraint on the independent variables.

As for comparing derivatives, I wouldn't do that. It merely confuses the issue. Set the derivatives equal to zero and find the set of points (x,y,z) that satisfy those conditions. In addition, you'll need to evaluate the function f on the boundaries of the regions in which you're looking.