Find the values of *x*,*y*,*z*≥0 that maximize the value of *f*(*x*,*y*,*z*)=*xyz*, subject to the constraint *ax*+*by*+*cz*=*d*, where *a*, *b*, *c* and *d* are positive real numbers.

This is a maximization subject to constraint problem, and is thus best approached by the method of Lagrange multipliers. Thus, we find the stationary points of the function:

(where we have chosen –*λ* rather than +*λ* to make some later algebra easier).

Taking the partial derivatives of *F*, we see

Setting the first three of these equal to zero, we obtain

And setting the fourth equal to zero gives us our constraint , as expected.

Now, multiplying those three equations we obtained from setting the partial derivatives equal to zero, we obtain

and taking the square root of both sides (with *λ*≥0),

Dividing this by the first equation, we see

And analogously,

And thus,

and from our constraint , we see that

(from which we can solve for *λ*, but this is unnecessary).

Thus, the point which maximizes *f*(*x*,*y*,*z*)=*xyz* is , with the maximum value thus being .

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Tags: Lagrange Multiplier, Math, Monday Math

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