40 2. The First Variation a x 1 x 2 b x Figure 2.5. A nonnegative bump To be able to apply this fundamental lemma of the calculus of variations, we must be sure that M(x) = ∂f ∂y − d dx ∂f ∂y (2.54) is continuous on the closed interval [a, b]. If we apply the chain rule, we may rewrite the right-hand side of this last equation in the ultra- differentiated form M(x) = ∂f ∂y − ∂fy ∂x dx dx − ∂fy ∂y dy dx − ∂fy ∂y dy dx (2.55) = fy − fy x − fy y y − fy y y . To obtain the Euler–Lagrange equation using Lagrange’s simplifica- tion, we must therefore make the additional assumption that ˆ (x) ∈ C[a, b] or that ˆ(x) ∈ C2[a, b]. Having made (or, more honestly, having been forced into) the assumption that ˆ(x) ∈ C2 [a, b], we can now state the following nec- essary condition for a relative maximum or minimum: η ()

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