Positive Hessian of a layered travel-time functional

For three unit-height layers with positive speeds $u,v,w$, the straight-segment travel time is $t=\sqrt{1+x_1^2}/u+\sqrt{1+(x_2-x_1{)}^2}/v+\sqrt{1+(X-x_2{)}^2}/w$. Its Hessian matrix has form $\left(\begin{array}{cc}p+q& -q\\ -q& q+r\end{array}\right)$ with positive $p,q,r$. Its leading minor is positive and its determinant is $pq+pr+qr>0$. Hence this time function is strictly convex, and its stationary path is the unique minimum.