Weak and strong extremum

Summary

In the calculus of variations a local minimum depends on what "nearby" means for functions. A strong local minimum beats every admissible function whose values are close. A weak local minimum only beats those whose values and slopes are both close. Every strong local extremum is a weak one, but not conversely. The Euler–Lagrange equation and the Legendre condition are necessary for weak extrema, while the Weierstrass excess function gives a further necessary condition for strong ones.