for my functional
$S[y]=\int\limits^{1}_{0}dx(y')^{n}e^{y}\,,$ $y(0)=1,y(1)=A>1$
i have found the stationary path to be
$y=n$ln$(cx+e^{1/n})$, where $c=e^{A/n}-e^{1/n}$
i have then found the Jacobi equation to be
$2c(cx+d)u'+(cx+d)^2u''=0$
which ive solved to get
$u=\frac{d}{c}-\frac{-d^2}{c}(cx+d)^{-1}$
which,if this is correct gives me the only solution of $u=0$ is given by $x=0$
so am i correct to say that this means the stationary path is minimal,since $\frac{\partial^2}{\partial y'^{2}}[(y')^{n}e^{y}]$ is positive?