Inspired by Harsh Shrivastava

Algebra Level 5

Let \(M\) be the maximal value of \(xy+2yz+3xz\), subject to the constraint \(x^2+y^2+z^2=2\), where \(x,y\) and \(z\) are real numbers. If \(M^3=aM+b\) for integers \(a\) and \(b\), enter \(a+b\).

Hint: Use the Spectral Theorem from Linear Algebra, Theorem 3.1, (or, equivalently, Lagrange multipliers)

Bonus question: If \(m\) is the minimal value, write \(m^3=cm+d\).



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