*z* is a constant, yes. *e* is also an arbitrary constant, not Euler's number. The only relationship between the constants is

b > d + z/f

I'm starting to wonder if I've missed something from context or something strange like that.

At this point I'd settle for proof that psi'(0) is negative iff z/(f^2) > e, which is the conclusion the authors draw, and as long as that's correct I think everything else hangs together.

I'm just thoroughly confused, because it seems highly unlikely that the authors would be wrong, but I can't see how they could be right either...

Cool, then I think we can get it

Inserting y=0 in the denominator makes it: f(b-d)-z = f( b - d - z/f), but b > d+z/f => b - d - z/f > 0, i.e. it can't be zero and (especially squared) it will be positive and therefore not affect the sign of the derivative.

So we just have to look at the nominator of the fraction derivative: n'(y) * d(y) - d'(y)*n(y)

I get

n'(y) = (d+ey) + (f+y)*e

d'(y) = (b-d-ey) +(f+y)*(-e)

Inserting 0 as y in these and slowly lifting the (negative) parentheses ends me with a lot of cancelling parts, so only bef

^{2} - bz is left

I.e. Phi'(0) = a*(bef

^{2} - bz) / f

^{2}* 'something positive' = -ab*(z/f

^{2}-e) / 'something positive'.

I.e. The sign of the derivative at y=0 is indeed negative iff z/f

^{2}-e > 0 <=> z/f

^{2} > e

Let me know if you want to compare how I lift the parentheses.

(It would seem that there is an typo in the book, as the equation for phi'(0) would be correct if the denominator was squared.)