Question 1199733
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See the link 
https://www.quora.com/If-u-v-w-x-y-15-then-what-is-the-maximum-value-of-uvx-uvy-uwx-uwy


the answer by Daniel Claydon.  Below I copy-pasted from there.


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Notice that

uvx + uvy + uwx + uwy = u(x+y)(v+w).

If  u, v, w, x, y are allowed to be negative, there is no largest value, since, for example, 
one could let  u, x, y be arbitrarily “large” negative numbers (so  u(x+y) is positive), 
then  v+w is a large positive number and the whole product can be as large as you like.

If they are restricted to positive real numbers, then from AM-GM, we have

    {{{u+(x+y)+(v+w)/3}}} >= {{{root(3,u(x+y)(v+w))}}}


Using the imposed condition to the sum, the left side is just  5, so we have found

    {{{u*(x+y)*(v+w)}}}} <= {{{5^3}}} = 125.


The maximum value is  125, and occurs if, and only if,  u = x+y = v+w = 5.
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