If #A, B, C# and #D# are positive numbers such that #A + 2B + 3C + 4D = 8#, then what is the maximum value of #ABCD#?

1 Answer
NickTheTurtle
May 13, 2017

#ABCD≤2/3#

Explanation:

The arithmetic mean-geometric mean inequality states that, for positive real numbers, the arithmetic mean of these numbers is always greater than or equal to the geometric mean of these numbers.

Thus, the arithmetic mean of #A,2B,3C,4D# is greater than or equal to the geometric mean of #A,2B,3C,4D#, or #(A+2B+3C+4D)/4≥root(4)(24ABCD)#. Since #A+2B+3C+4D=8#, #8/4≥root(4)(24ABCD)#, or #ABCD≤2/3#.

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