Cool Inequality 6 - Making it Really Tough

Algebra Level 5

Let the real numbers \(a,b,c,d\) satisfy the relation \(a+b+c+d=6\) and \(a^{2}+b^{2}+c^{2}+d^{2}=12\). Then find the absolute value of difference of the Maximum and minimum value of

\[\large{4\left( { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }+{ d }^{ 3 } \right) -\left( { a }^{ 4 }+{ b }^{ 4 }+{ c }^{ 4 }+{ d }^{ 4 } \right) }\]

Bonus:-Try to solve it without using calculus

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