= Solution
At a smooth interior stationary point, define $D=t_{11}t_{22}-t_{12}^2$. The <second-derivative test> gives a minimum for $D>0,t_{11}>0$, a maximum for $D>0,t_{11}<0$, and a <saddle point> for $D<0$; $D=0$ is inconclusive.
For the stated time function, put
$$
p=\frac1{u(1+x_1^2)^{3/2}},\quad q=\frac1{v(1+(x_2-x_1)^2)^{3/2}},\quad r=\frac1{w(1+(X-x_2)^2)^{3/2}}.
$$
Positive physical speeds make all three coefficients positive. The <positive Hessian of a layered travel-time functional> here is
$$
H=\begin{pmatrix}p+q&-q\\-q&q+r\end{pmatrix},\qquad \boxed{t_{11}=p+q>0,\quad D=pq+pr+qr>0.}
$$
Thus the stationary path is a strict minimum, not merely a candidate of unspecified type. In fact the same positivity holds for all finite crossing positions, so the time function is strictly convex and the minimum is unique.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-layered-ray-travel-time.png]
{title=Minimum travel time through three constant-speed layers and the crossing-position time surface}
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