In PRML, formula (4.130) implies that $A>0$, and the context says that the second derivative of $f(z)$ should be negative, where: $$ A=-\frac{d^2}{dz^2}lnf(z)\big|_{z=z_0} $$ However, by deriving the right side, we have: $$ A=\frac{f'(z)^2-f''(z)f(z)}{f^2(z)}\Big|_{z=z_0} $$ and it seems that $f''(z_0)$ doesn't need to be negative; hence $z_0$ may not be the maximum.
In conclusion, I think $z_0$ doesn't need to be the (local)maximum, or am I wrong?