Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 9 / ii / a / Solution

The integrand $f=n(x)\sqrt{1+(y^{\prime }{)}^2}$ has no explicit dependence on $y$, so part (i) givesFor positive refractive index and a regular graph with $n(x)>|c|$, squaring and then restoring the sign of $y^{\prime }$ yieldsIt is the continuous-medium version of Snell's law, expressing conserved horizontal optical momentum. The Euler-Lagrange equation establishes stationarity; on a monotone-height segment, convexity of the integrand in $y^{\prime }$ also gives a minimum among graphs with fixed endpoints.