Can Derivative Work?

Algebra Level 5

\[ \large{\begin{cases} x+y+z=4 \\ x^2+y^2+z^2 =6 \end{cases}} \]

Real numbers \(x,y\) and \(z\) satisfy the system of equations above.

Let \(P =x^3+y^3+z^3\). If the difference between the maximum value of \(P \) and the minimum value of \(P \) can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

Bonus: Generalize the result for \(x+y+z=m\) and \(x^2+y^2+z^2=n\).

Source: Grade 10 Gifted Math Students Selection Test, University of Education, HCMC, Vietnam.

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