= Solution
For fixed-endpoint variations, the <second variation> is
$$
q(\varphi)=\delta^2I(\bar u)(\varphi,\varphi)=\left.\frac{d^2}{dt^2}I(\bar u+t\varphi)\right|_{t=0}=\int_a^b(p\varphi'^2+2r\varphi\varphi'+s\varphi^2)dx,
$$
with $p=f_{vv}$, $r=f_{uv}$, $s=f_{uu}$ evaluated along $(x,\bar u,\bar u')$. The <Jacobi equation>, written without differentiating $r$ separately, is
$$
\boxed{-(pz'+rz)'+rz'+sz=0.}
$$
When $r$ is differentiable this is $-(pz')'+(s-r')z=0$.
<Positive definiteness> means $q(\varphi)>0$ for every nonzero admissible variation. Uniform or <coercive positivity of a second variation> is the stronger assertion that $q(\varphi)\geq c\int\varphi'^2$ for some fixed $c>0$. On a fixed-endpoint interval this is equivalent to a positive lower bound in the $W^{1,2}$ norm.
\b[The strict Legendre implication uses the uniform interpretation of positivity.] If $q\geq c\int\varphi'^2$, localized rapidly oscillating variations show $p(x)\geq c$: otherwise support them in an interval where $p-c<0$, so the <derivative> term of $q-c\int\varphi'^2$ is negative of order frequency squared, overwhelming its mixed and zero-order terms. Continuity extends the bound to the endpoints, giving $p>0$ on $[a,b]$. Under mere <positive definiteness> this strict conclusion is false: $f=u^2$ at $\bar u=0$ gives $q=2\int\varphi^2>0$ but $p=0$. In that weaker convention only the ordinary <Legendre necessary condition> $p\geq0$ follows. This distinction makes the implication precise.
Under the stated regular condition $p>0$, a <conjugate point> $c\in(a,b]$ to $a$ is an endpoint admitting a nonzero Jacobi solution with $z(a)=z(c)=0$. The disconjugacy form of the <Jacobi positivity criterion> says that no such point in $(a,b]$ exists if and only if there is a Jacobi solution strictly positive on all of $[a,b]$.
For the required identity, let that positive solution be $z$ and write $\varphi=zh$. Set $A=pzz'+rz^2$. The <Jacobi equation> gives
$$
A'=pz'^2+2rzz'+sz^2.
$$
Expanding the second-variation integrand therefore yields
$$
p\varphi'^2+2r\varphi\varphi'+s\varphi^2=pz^2h'^2+(Ah^2)'.
$$
For compactly supported $\varphi$, the <boundary term> vanishes, proving
$$
\boxed{\delta^2I(\bar u)(\varphi,\varphi)=\int_a^bpz^2\left[\left(\frac\varphi z\right)'\right]^2dx.}
$$
This is the <Jacobi identity for the second variation>. The divergence form is also appropriate under the original $C^1$ regularity; with $p>0$, the <Euler-Lagrange equation> gives the usual further regularity along the extremal.
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