What is the process for finding the maximum and minimum of a discrete model?

[Dustinsfl]

To find the max and min of a discrete model, I solve

$$
\frac{df}{dN_t} =0\Rightarrow N_m
$$
Then $N_{\text{max}} = f(N_m)$, correct?

Because whenever I try to solve

$$
N_{t+1} =\frac{(1+r)N_t}{1+rN_t}
$$

It doesn't work.

The derivative is
$$
\frac{1+r}{(1+rN_t)^2}=0
$$

 

[Aaron Bex]

so
$$
N_m=\frac{1}{r}
$$
and
$$
N_{max} = f(\frac{1}{r})
$$