As per request from Comrade Pi Han Goh

Algebra Level 5

\[\begin{eqnarray} f_a(x,y,z)&=&ax^2+(a+1)y^2+(a+2)z^2 \\ && -4xy-4yz+2x-3y+4z-5 \end{eqnarray}\]

Find the smallest real number \(a\) such that the function \(f_a(x,y,z)\) has a (global) minimum.


Motivation.

The Topic is Algebra; calculus solutions will be frowned upon.
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