Discrete maximum principle for discretized ODE

I discretized the following ODE using central finite differences for 1st and 2nd derivatives: $$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$

The function $$f$$ here satisfies a Lipschitz condition (i.e. $$|f(u_1)-f(u_2)| \le K|u_1-u_2|$$) for some $$K\ge 0$$. So I got

$$\left(1-\frac{hb}{2}\right)u_{i+1}-2u_i+\left(1+\frac{hb}{2}\right)u_{i-1}=f(u_i)$$

Now I want to see if this equation satisfies a discrete maximum principle.

If my understanding is correct then I need to show that

$$\left(1-\frac{hb}{2}\right)u_{i+1}-2u_i+\left(1+\frac{hb}{2}\right)u_{i-1}\le 0$$or $$\left(1-\frac{hb}{2}\right)u_{i+1}-2u_i+\left(1+\frac{hb}{2}\right)u_{i-1}\ge 0$$

for all $$x\in (0,1)$$.

But I don't see how to approach this problem using the Lipschitz condition given for $$f$$ or otherwise. Would appreciate some hints.