Hmm squares

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 $f(b, c)=M$