While learning differential equations, I was reading some notes, and it was mentioned that for Dirichlet BVP

$$x'' = f (t, x), \quad x(0) = 0 = x(1).$$ Suppose $f : [0, 1] \times \mathbb{R}\to \mathbb{R}$ is continuous and there is a constant $R > 0$ such that $f (t, R) \ge 0$, $f (t, −R)\leq 0$, for all $t \in [0, 1]$. It can be shown that there is at least one solution, but the proof is missing.

Can someone please help me out with this?

How to find such an constant $R$, for some problem say, $$x'' = x^3 + t,\quad x(0) = 0 = x(1)$$ to show that this has at least one solution?

Regards, Salil