# Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary

Suppose we want to make an integral $$S$$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $$y_1\left(x_1\right) = \tilde{y}_1, \dots, y_n(x_1) = \tilde{y}_n, y_1\left(x_2\right) = \bar{y}_1, \dots, y_n(x_2) = \bar{y}_n$$. Is it necessary, sufficient, or necessary and sufficient for the $$y_1(x), \dots, y_n(x)$$ to satisfy the $$n$$ Euler-Lagrange equations, $$\frac{\partial f}{\partial y_1} = \frac{d}{dx}\frac{\partial f}{\partial y'_1}, \dots,\frac{\partial f}{\partial y_n} = \frac{d}{dx}\frac{\partial f}{\partial y'_n},$$ in order to make $$S$$ stationary?