Sum of n numbers constant maximum product
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+y+z=73
and
x,y,z∈
I
+
x,y,z∈I+
. Show that the maximum value of
xyz=14400
xyz=14400
.
Now using the AM-GM inequality one can establish that the maximum value of the product for real values of the variable would be
≈14408.037037
≈14408.037037
. However we have the added clause that x,y,z must be positive integers.
Intuitively, it seems that whenever
∑i=1n|xn−
xi
|
∑i=1n|xn−xi|
is minimised for the general case, the product
x
1
x2
....
xn
x1x2....xn
is maximized. So 24,24,25 is the case in which the value of the above expression for positive integral values of the variables is minimized and hence the product is maximized. But I would like a proof of the above result.
and
x,y,z∈
I
+
x,y,z∈I+
. Show that the maximum value of
xyz=14400
xyz=14400
.
Now using the AM-GM inequality one can establish that the maximum value of the product for real values of the variable would be
≈14408.037037
≈14408.037037
. However we have the added clause that x,y,z must be positive integers.
Intuitively, it seems that whenever
∑i=1n|xn−
xi
|
∑i=1n|xn−xi|
is minimised for the general case, the product
x
1
x2
....
xn
x1x2....xn
is maximized. So 24,24,25 is the case in which the value of the above expression for positive integral values of the variables is minimized and hence the product is maximized. But I would like a proof of the above result.
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