= Solution
With coefficients evaluated along $(x,\bar u,\bar u')$, put $p=f_{vv}$, $r=f_{uv}$ and $s=f_{uu}$. The <second variation> is
$$
q(\phi)=\delta^2I(\bar u)[\phi,\phi]=\int_a^b(p\phi'^2+2r\phi\phi'+s\phi^2)dx.
$$
The <Jacobi equation> in divergence form is $-(pz'+rz)'+rz'+sz=0$; if $r$ is differentiable it is $-(pz')'+(s-r')z=0$. A <conjugate point> $c>a$ admits a nonzero Jacobi solution vanishing at $a,c$.
There are two positivity conventions worth distinguishing. Strict pointwise positivity means $q(\phi)>0$ for each nonzero fixed-endpoint variation. Coercive positivity means $q(\phi)\geq c\int\phi'^2$ with one $c>0$ for all such variations; the endpoint Poincaré estimate makes this equivalent to a positive lower bound in the full $H^1$ <norm>. Under the regular Legendre hypothesis $p>0$, the necessary and sufficient Jacobi criterion is \b[no <conjugate point> in $(a,b]$]; in that regular case it gives both strict and coercive positivity. Equivalently the <Jacobi equation> has a solution positive on $[a,b]$, and the square identity
$$
q(\phi)=\int_a^bpz^2[(\phi/z)']^2dx
$$
gives positivity and a uniform derivative lower bound by the positive minimum of $z$ and boundedness of $z,z'$. Conversely a conjugate Jacobi field, extended by zero beyond its second zero, is a nonzero Lipschitz variation with $q=0$, contradicting either positivity convention.
Coercivity itself forces $p>0$: local high-frequency variations make the leading coefficient at least the coercivity constant. Mere strict positivity need not force it, as $q(\phi)=\int\phi^2$ shows. If that weaker convention is intended without regularity, a complete criterion is $p\geq0$, no <conjugate point> for every regularized form $q_\eta=q+\eta\int\phi'^2$ with $\eta>0$, and no nonzero zero-endpoint Lipschitz <weak solution> of the original <Jacobi equation>. The regularized criteria imply $q_\eta\geq0$, hence $q\geq0$ by passage to the limit. For a nonnegative quadratic form, $q(\phi)=0$ forces all mixed terms with test variations to vanish, giving exactly the weak Jacobi kernel; excluding it gives strict positivity. Necessity follows in reverse. This specifies the degenerate case without imposing an unstated strict Legendre condition.
For weak local minimality, sufficient conditions are the <Euler-Lagrange equation> and a coercively positive <second variation> with the stated smoothness. Taylor's remainder is bounded by $\varepsilon\|\phi\|_{H^1}^2$ for small $W^{1,\infty}$ perturbations, so the positive quadratic term dominates it. These are the classical \b[weak minimum sufficient conditions].
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