  We introduce the nonlinear generalized Collatz-Wielandt formula $$ \lambda^*=
\sup_{x\in Q}\min_{i:h_i(x) \neq 0} \frac{g_i(x)}{ h_i(x)}, ~~Q \subset
\mathbb{R}^n,$$ and prove that its solution $(x^*,\lambda^*)$ yields the
maximal saddle-node bifurcation for systems of equations of the form:
$g(x)-\lambda h(x)=0, ~~x \in Q$. Using this we introduce a simply verifiable
criterion for the detection of saddle-node bifurcations of a given system of
equations. We apply this criterion to prove the existence of the maximal
saddle-node bifurcations for finite-difference approximations of nonlinear
partial differential equations and for the system of power flow equations.
