= Solution
Within the requested first-quadrant square, the <stationary points> visible in the <contour plot> are the corner minimum $(0,0)$ and the saddle $(1,1)$. Near the origin, $g\simeq4(x^2+y^2)$, so the low positive contours look like quarter-circles. Along either coordinate axis, $g=4t^2+t^4$ rises monotonically. Along the diagonal, $g(t,t)=8t^2-4t^4$ rises to $4$ at $t=1$, returns to zero at $t=\sqrt2$, then becomes negative.
The saddle contour $g=4$ crosses at $(1,1)$ and separates the central positive contours from the outer diagonal region. Negative contours occur toward $(2,2)$, where $g=-32$, while the axis corners $(2,0)$ and $(0,2)$ have $g=32$. These checks determine both the contour ordering and the change of connectivity at the saddle; the figure shows these level sets.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-quartic-contours.png]
{title=Contours of the quartic function in the first-quadrant square, with the saddle level highlighted}
Back to article page
