Hmm squares

Algebra Level 5

Suppose \(b, c\) are any positive reals such that \(b^{2}+c^{2}=1\). Then, let \(M\) the largest possible value of

\[f(b, c)=b(1-b^{2})(4c^{2}-3)+\frac {b}{4}\]

The smallest possible value of \(c\) such that equality occurs can be expressed as \(\displaystyle \sqrt {\frac {x-\sqrt {y}}{z}}\) where \(x, y, z\) are positive integers. Find the smallest possible value of \(x+y+z\).

  • Equality occurs meaning that for the value of \(c\), there exists \(b\) satisfying the above conditions such that \(f(b, c)=M\)

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