Inspired by Brian Charlesworth, again

Algebra Level 5

\[10xy+12xz+14yz\]

Let \(M\) and \(m\) be the maximum and the minimum of the expression above if \(x^2+y^2+z^2=1\), where \(x,y\) and \(z\) are real numbers. Find \(M^2+Mm+m^2\).

Bonus question: For \(2axy+2bxz+2cyz\), with the same constraint, express \(M^2+Mm+m^2\) in terms of \(a,b\) and \(c\).


Inspiration

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