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Fox states in Section 2.4, pg. 38, that

"Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still vanish since $t(a)=t(b)=0$ by hypothesis."

How does this make sense? The quotient would become $0/0$ in such a case, which is indeterminate, not vanishing.

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  • $\begingroup$ @Cesareo Any ideas? $\endgroup$ – A. Hendry Mar 21 at 23:41
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I'm just going to assume the Jacobi Necessary Condition is only looking at the internal interval $(a,b)$ and call it a day.

EDIT 3-25-19:

This is true (the necessary condition looks only at the open interval). Hence, Fox's statement is ultimately incorrect IFF $u$ were to equal $0$ at any point, but this is not possible (see Difficulty Understanding Sufficient Conditions for Weak Extrema in Calculus of Variations).

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