How does one maximize the following equation? ( a &#x2212;<!-- − --> c ) ( b

cambrassk3

cambrassk3

Answered question

2022-07-07

How does one maximize the following equation? ( a c ) ( b c ) ( a b ) given that n c < b < a n for some value of n
So far, I have attempted to do this using AM-GM and got 64 27 n 3 but I don't think this is correct as some of the values of a, b, and c can be negative and when I tried to find the actual values which satisfied the max value found it gave me a contradiction.

Answer & Explanation

Jamiya Costa

Jamiya Costa

Beginner2022-07-08Added 18 answers

2 n ( a c ) = ( a b ) + ( b c ).
All terms in parenthesis are non-negative, and AM-GM applies.

Hence, conclude that
( a c ) ( a b ) ( b c ) 2 n × n × n = 2 n 3
Clearly, this can be achieved with a = n , b = 0 , c = n.

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