How find this Geometry in space find this $|BB'|+|CC'|$ range? Question:

In  Space,let $\Delta ABC$ such $\angle B=90,\angle A=60 $ ,and $|AC|=4$;

and the point  $A$ is  on plane $\alpha$,and $M$ is $BC$ midpoint,if $BB'\perp \alpha,CC'\perp \alpha$;
and $B',C'$ is on the plane $\alpha$,and such $\angle B'AC'=90$.
find the

$\sup{(|BB'|+|CC'|)}$ and $\inf{(|BB'|+|CC'|)}$


This quetion aswer is $(2,\dfrac{5}{2})$,in other words 

$\sup{(|BB'|+|CC'|)}=\dfrac{5}{2}$ and $\inf{(|BB'|+|CC'|)}=2$

**My try:we have **

$$BC^2=AC^2-AB^2=12$$ and let $$||BB'|=x,|CC'|=y,\Longrightarrow x<2,y<4,y\ge x$$
  so
  $$AB'^2=4-x^2,AC'^2=16-y^2$$
  since $$|B'C'|^2\le |BC|^2\Longrightarrow 4-x^2+16-y^2\ge 12\Longrightarrow x^2+y^2\ge 8$$
  then
  $$|BB'|+|CC'|=x+y$$ 
  and I can't have this answer.Thank you 

 A: First, we observe that $\triangle ABC$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle, so that $|AC| = 4$ implies $|AB| = 2$ and $|BC| = 2 \sqrt{3}$.
Now, exploiting various right triangles:
$$\begin{align}
|BC|^2 &= |B^\prime C^\prime|^2 + \left(\; |BB^\prime| - |CC^\prime| \;\right)^2 \\
&= \left( \; |AB^\prime|^2 + |AC^\prime|^2 \; \right) + \left(\; |BB^\prime|^2 + |CC^\prime|^2 - 2 |BB^\prime||CC^\prime| \;\right) \\
&= \left(\; |AB^\prime|^2 + |BB^\prime|^2 \;\right) + \left(\; |AC^\prime|^2 + |CC^\prime|^2 \;\right) - 2 |BB^\prime||CC^\prime| \\
&= |AB|^2 + |AC|^2 - 2 |BB^\prime||CC^\prime| \\[6pt]
\implies |BB^\prime||CC^\prime|&= \frac{1}{2}\left(\;|AB|^2+|AC|^2-|BC|^2\;\right) = 4
\end{align}$$
Writing $b := |BB^\prime|$ and $c := |CC^\prime|$, you're considering $b+c$ subject to the conditions
$$0 \leq b \leq |AB| = 2 \qquad 0 \leq c \leq |AC| = 4 \qquad b c = 4$$
which means you're really considering
$$f(b) := b + \frac{4}{b}$$
subject to the condition $$1 \leq b \leq 2$$
(The maximum value of $c$ is $4$, so that the minimum value of $b$ must be $4/4 = 1$.) Since $f$ decreases over the given domain, the extrema occur at the endpoints:
$$\inf f(b) = f(2) = 4 \qquad\qquad \sup f(b) = f(1) = 5$$
Hmmmm ... These values are exactly twice what you're supposed to get, but I don't see my error.
