Let $y_0 \in C^2([a,b])$ be an extremal of $I[y] = \int_a^b L(x,y,y')\, dx$ satisfying the strengthened Legendre condition $P(x) = \partial_{y'y'} L(x,y_0,y_0') > 0$ on $[a,b]$. If $y_0$ gives a (weak) local minimum of $I$, then there is no point conjugate to $a$ in the open interval $(a,b)$, i.e., $a^* \notin (a,b)$.