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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