2.3. Lagrange’s approach 43 Now, consider the variation η(x) defined by the equation η(x) = x a [M(u) − μ] du . (2.68) It is easy to see that η(x) ∈ C1[a, b]. The function η(x) also vanishes at x = a and x = b. It is clearly an admissible variation. Moreover, η (x) = M(x) − μ . (2.69) By hypothesis, b a M(x) η (x) dx = b a M(x) [M(x) − μ] dx = 0 . (2.70) Also, b a M(x) [M(x) − μ] dx − μ b a [M(x) − μ] dx = 0 . (2.71) But, this last equation may be rewritten b a [M(x) − μ]2 dx = 0 . (2.72) Let x0 ∈ [a, b] be a point where M(x) is continuous. If M(x0) = μ, then there would have to exist a subinterval about x = x0 on which M(x) = μ. But this is clearly impossible in light of our last displayed equation. Thus M(x) = μ at all points of continuity. It follows that M(x) is constant for all x ∈ [a, b]. ♣ We now wish to apply this lemma to necessary condition (2.62), b a ⎛ ⎝ ∂f ∂y − x a ∂f ∂y du⎠ ⎞ ˆ ˆ η (x) dx = 0 . (2.73)

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