To the best of my knowledge, the function that minimizes of the integral posed in calculus of variations must also satisfy the Euler-Lagrange equations. In other words, the Euler equations are a necessary condition for finding a minimizer.
One thing I do not understand, is, why do I see everyone treating it as if it were a sufficient condition to show optimality? I am solving some example problems in calculus of variations, and I am following the problems here: http://matematika.cuni.cz/dl/pyrih/variationProblems/variationProblems.pdf example
For example, look at problem 1.1 on page 2, which says:
Using the Euler equation find the extremals for the following functional: $\int_{a}^{b}12xy(x)+(\frac{\partial y(x)}{\partial x})^2dx$
then, as I am looking at the solution which is immediately below, it says "finally obtain the Euler equation for our functional"...and proceeds to solve the Euler equation as a means to solve the minimization problem to obtain the answer.
I do not understand why this is allowed, if the Euler equation is a necessary condition, but it is not sufficient.
Is the Euler equation necessary and sufficient? or just necessary?
An example of a resource that explicitly says it, says "Euler-Lagrange Equation and is a necessary, but not sufficient, condition for an extremal function", is here: http://www.maths.manchester.ac.uk/~wparnell/MT34032/34032_CalcVar