Maxima and Minima of a Function of Several Variables.

Jun 2010
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0
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

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Jun 2010
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Are you trying to find the extrema over some region of R^3?
 

HallsofIvy

MHF Helper
Apr 2005
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Take the partial derivatives and set them equal to 0. Solve those equations for x, y, and z. Can you find the partial derivatives?
 
Feb 2010
168
20
Columbus, Ohio, USA
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.
 
Jun 2010
27
0
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
Jun 2010
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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.