A constrained optimum cannot improve by moving along the constraint.
For f=ax+by on a radius-R circle, Cauchy–Schwarz gives |f|≤R√(a²+b²). When (a,b)≠0, equality occurs at ±R(a,b)/√(a²+b²), with λ=±√(a²+b²)/(2R). The constraint gradient is nonzero because R>0. If a=b=0 every point is optimal.
Parallel gradients identify candidates under regularity conditions, not automatically a global maximum in every problem.