An exact multiplicity result of positive solutions for the boundary value problems u′′+λa(t)f(u)=0, t∈(0,1), u′(0)=0, u(1)=0 is achieved, where λ is a positive parameter. Here the function f:[0,∞)→[0,∞) is C2 and satisfies f(0)=f(s)=0, f(u)>0 for u∈(0,s)∪(s,∞) for some s∈(0,∞). Moreover, f is asymptotically linear and f″ can change sign only once. The weight function a:[0,1]→(0,∞) is C2 and satisfies a′(t)<0, 3(a′(t))2<2a(t)a′′(t) for t∈[0,1]. Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for λ lying in various intervals in R. Moreover, we indicate how to extend the result to the general case.