We consider the continua of positive solutions of the one parameter problem $$-u''(x)=\lambda f(u(x)),\ \ \ \ x\in(0,1),$$ $$u(0)=u(1)=0,$$ where $f\in C^1[0,\infty)$, $f(0)=0$ and there exist $0<s_0<s_1<s_2<s_3$ such that $f(s_0)=f(s_1)=f(s_2)=f(s_3)=0$ and $f(s)<0$ for $s\in(s_0,s_1)\cup(s_2,s_3)$, $f(s)<0$ for $s\in(s_1,s_2)$.
We can get the existence of the continua $\Sigma_0$, $\Sigma_1$ and $\Sigma_\infty$, and the sharp lower bounds on the $C$-norms of solutions on the continua $\Sigma_1$ and $\Sigma_\infty$, namely:
(i) If $(u, \lambda)\in\Sigma_1$ then $||u||_{\infty}=\underset{x\in(0,1)}\max u(x)\geq r_1$, where $r_1\in(s_1,s_2)$ is given by $\int_{s_0}^{s_1}f(s)ds=0$;
(ii) If $(u, \lambda)\in_{\infty}$ then $||u||_{\infty}\geq r_{\infty}$, where $r_{\infty}>s_3$ is given by $\int_{s_2}^{r_{\infty}}f(s)ds=0$.
How can I get the fact (i) and (ii)?