Why is this a fake proof? Cauchy’s Mean Value Theorem

Question

We are assuming that g and f satisfy the mean value theorem and apply it to each, then divide f'(z) / g'(z), Why is this a fake proof?

As f and g satisfy the hypotheses of the mean value theorem we apply it and we have that there exists z that belongs to the interval (a, b)
such that $f '(z) = \frac{f (b) -f (a)}{ b- a}$ and
$g '(z) = \frac{g (b) -g (a)}{ b- a}$ then we divide $\frac{f '(z )} { g '(z)}$ and we have the proof of Cauchy's mean value cause this is equal to

$\frac{f '(z )} { g '(z)} = \frac{f (b) -f (a)}{ b- a} / \frac{f (b) -f (a)}{ (b- a)} = \frac{f(b)-f(a)}{g(b)-g(a)}$

Answer 1

It's a fake proof because the $z_1$ such that $f'(z_1) = \frac{f(a) - f(b)}{a - b}$ and the $z_2$ such that $g'(z_2) = \frac{g(a) - g(b)}{a - b}$ are not necessarily the same $z$.