Stiffness extrema in proportionally loaded structures can be characterized through two seemingly disparate mathematical frameworks: Mang's variational criterion, formulated through the functional Ψ = (1/2!)δ²Π + (1/3!)δ³Π and its associated stationarity conditions (equivalently, the inflection condition d²χ₁/dλ² = 0 on a special eigenvalue function in FEM form), and sensitivity analysis quantifying how geometric stiffness Kᴳ depends on section stiffness Kᴱ via an amplification factor β. While both approaches successfully identify critical design points where stiffness exhibits local maxima or minima, their fundamental connection has remained elusive. This paper establishes a rigorous unification through the coupling matrix C = I + ∂Kᴳ/∂Kᴱ, which encapsulates the nonlinear interdependence between section and geometric stiffness contributions. We prove that three conditions are mathematically equivalent under stated regularity assumptions: (a) Mang's variational criterion for stiffness extrema (the inflection condition χ1 = 0), (b) det(C) = 0 (singularity of the coupling matrix), and (c) β reversal (the total stiffness sensitivity dKᵀ/dx vanishes, corresponding to 1 + β = 0). This equivalence not only provides deeper theoretical insight into the physical mechanisms governing stiffness evolution, but also enables a computationally efficient extremum prediction strategy: the β-field serves as a rapid detector to localize high-sensitivity regions, followed by targeted application of Mang's method only in these zones. Validation through eight representative structures-including arches, cable networks, portal frames, spatial trusses, and shells-demonstrates 6-11× computational speedup (average 8.2×) with prediction errors below 1-3% and β-χ1 correlation coefficients consistently exceeding 0.85. The unified framework bridges variational mechanics and sensitivity analysis, opening new avenues for stiffness-based design optimization.