Let $y_0 \in C^2([a,b])$ be an extremal of $I[y] = \int_a^b L(x,y,y')\, dx$. Suppose: 1. The strengthened Legendre condition holds: $P(x) = \partial_{y'y'} L(x,y_0,y_0') > 0$ on $[a,b]$. 2. There is no conjugate point of $a$ in the half-open interval $(a,b]$, i.e., $a^* > b$. Then $\delta^2 I[y_0; h] > 0$ for all nonzero $h \in C^1([a,b])$ vanishing at the endpoints, and $y_0$ gives a strict weak local minimum of $I$.