We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation
$$
-\left({u'}/{\sqrt{1+{u'}{^2}}}\right)'=f(t,u),
u(0)=0, u(1)=0,
$$
in connection with the changes of concavity of the function $f$. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.
