# 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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