<s>[INST] Simplify the expression and find the minimum value I want to simplify the expression
\begin{equation}
A = \frac{\sqrt{1-\sqrt{1-x^2}}\Big[\sqrt{(1+x)^3} + \sqrt{(1-x)^3}  \Big]}{2-\sqrt{1-\sqrt{1-x^2}}}
\end{equation}
and find the minimum value of $\displaystyle \sqrt{2}A+\frac{1}{x}$.
I can only simplify to this point
\begin{equation}
A = \frac{2x\Big( 2-\sqrt{1-x^2} \Big)}{2\sqrt{2}-\sqrt{1+x}+\sqrt{1-x}}.
\end{equation}
However, I am not sure if I can go any further. The next part of the question is to find the minimum value of $\displaystyle \sqrt{2}A+\frac{1}{x}$, which is obviously not so easy since I am not supposed to use differentiation to find optimal values.
 [/INST] Try this: $u = \sqrt{1+x}, v = \sqrt{1-x} \to \sqrt{1-x^2} = uv, 2 = u^2+v^2$, and consider $f(u,v) = ...$
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