How does one maximize the following equation? ( a − c ) ( b − c ) ( a − b ) given that − n ≤ c &lt; b &lt; a ≤ n for some value of nSo 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.

Question

How does one maximize the following equation?   (  a  −  c  )  (  b  −  c  )  (  a  −  b  ) given that   −  n  ≤  c  &amp;lt;  b  &amp;lt;  a  ≤  n for some value of   nSo far, I have attempted to do this using AM-GM and got       64    27        n    3   but I don&#039;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.

Jamiya Costa · Accepted Answer

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.

How does one maximize the following equation? ( a − c ) ( b

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