Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 12 / a / Solution

Let $D=1+x^2+y^2$. The gradient has components $f_x=[D-2x(x+y)]/D^2$ and $f_y=[D-2y(x+y)]/D^2$. At a critical point, subtraction gives $(x-y)(x+y)=0$. The case $x+y=0$ cannot make either numerator zero, so $x=y=t$. The remaining equation is $1+2t^2=4t^2$, givingThese are the only stationary points; the denominator is positive everywhere.