# 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 $$x^2+y^2+z^2=n$$.

×