Does Quadratic Forms Help?

Algebra Level 5

f(a,b,c,d)=1a2+1+1b2+1+1c2+1+1d2+1 f(a, b, c, d) = {\frac { 1 }{ { a }^{ 2 }+1 } +\frac { 1 }{ { b }^{ 2 }+1 } +\frac { 1 }{ { c }^{ 2 }+1 } +\frac { 1 }{ { d }^{ 2 }+1 } }

Let a,b,c,da,b,c,d to be non negative real numbers satisfying ab+ac+ad+bc+bd+cd=6ab+ac+ad+bc+bd+cd=6. Let minf(a,b,c,d)=M \min f(a, b, c, d) = M .

How many ordered quadruples of non-negative real numbers (a,b,c,d) (a, b, c, d) satisfy f(a,b,c,d)=M f(a,b,c,d) = M ?

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