Simple proof of the Lagrange multiplier rule for the fixed endpoint problem in the calculus of variations

A new simple proof of the Lagrange multiplier rule is presented in this paper. The approach used involves simple analytical techniques that are very easy to follow and does not involve theorems on imbeddability in a one-parameter family of curves or matrix-rank analysis as do most of the existing techniques. The proof is here developed for the fixed-endpoint problem in a three-dimensional space.