if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$

Question

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$

I don't know how can I start to solve this question . I hope you can help me in solving it or even giving me hints .

Answer 1

Hint: consider $(A+B+C)^2$, and look up the AM-GM inequality.