Summary: In the recent past many results have been established on non-negative solutions to boundary value problems of the form \(- u''(x)= \lambda f(u(x))\), \(0< x< 1\), \(u(0)= 0= u(1)\), where \(\lambda> 0\), \(f(0)< 0\) (semi-positone problems). In this paper, we discuss the existence of non-negative solutions when the boundary conditions are replaced by \(u(0)= 0= u(1)+ \alpha u'(1)\), where \(\alpha> 0\), via a quadrature method. In particular, we consider the case when \(f\) is superlinear. We also establish critical bounds for the bifurcation diagram of these non-negative solutions and study its evolution within these bounds as \(\alpha\) varies. We further discuss the existence of multiple positive solutions for ranges (which are independent of \(\alpha\)) of \(\lambda\). We discuss these results both for semi-positone as well as positone \((f(0)> 0)\) problems.