Introduction to the Calculus of Variations

In dealing with a function of a single variable, y = f (x), in the ordinary calculus, we often find it of use to determine the values of x for which the function y is a local maximum or a local minimum. By a local maximum at position x 1, we mean that f at position x in the neighborhood of x 1 is less than f (x 1) (see Fig. 2.1). Similarly for a local minimum of f to exist at position x 2 (see Fig. 2.1) we require that f (x) be larger than f (x 2) for all values of x in the neighborhood of x 2. The values of x in the neighborhood of x 1 or x 2 may be called the admissible values of x relative to which x 1 or x 2 is a maximum or minimum position.