Calculus/Complex Numbers exercise question.

Question

Let $f$ be an increasing function in $\mathbb{R}$ and $f(-1)>0$.
Let $z=\frac{f(-1)f(0)}{2}+ \frac{4\sqrt{3}}{f(1)}i$ with $|z|=\frac{z}{2}(1-\sqrt{3}i)$.
Show that:
i) $f(-1)f(0)f(1)=8$
ii) $2Re(z)=|z|$
iii) $f(-1)<2<f(1)$
iv)$ \exists$ $f^{-1}$ and $-1 < f^{-1}(\sqrt{|z|}) < 0 $

I'm ok with i, ii, iv but I'm having difficulty with number iii. Any hint is gladly accepted.

Answer 1

If $f(-1)f(0)f(1) = 8$ and $f$ is increasing, can $f(-1) \ge 2$ be true? How about $f(1)\le 2$?