I would like to understand Smale's mean value problem. I am sure I'm misunderstanding it, as my misinterpretation seems easy to disprove. Would anyone be able to help point out what I am missing?
The problem is this: for a given complex polynomial $p$ of degree $d\ge2$ and a complex number $z$, is there a critical point $c$ of $p$ (i.e. $p^\prime (c)=0$ such that
$$\left|\frac{p(z)-p(c)}{z-c}\right| \leq \left|p^{\prime}(z)\right| \, ?$$
I am misunderstanding something, because this seems like an obvious "no". I can choose
$$p(z) = \frac{1}{3}z^3 - (b+c)z^2 +bcz$$
with
$$p^\prime(z)=z^2-2(b+c)+bc = (z-b)(z-c)$$
so that both $b$ and $c$ are critical points, and
$$p(b) = -\frac{2}{3}b^3 \, , \quad p(c) = -\frac{2}{3}c^3$$
Then it's easy to choose $b,c$ (say, $b=1,c=0$) such that
$$\left|\frac{p(b)-p(c)}{b-c}\right| > \left|p^{\prime}(c)\right|=0 $$
What am I misunderstanding here? Though I would expect this problem to be out of reach for me right now, I would at least like to understand it.