# Cyclic descent

$LaTeX: f(x, y) = (y - x^2)(y - 2x^2) \; \mbox{ at } \; (x, y) = (0,0)$.
$LaTeX: f(0, y)$ has a minimum at $LaTeX: y=0$, and $LaTeX: f(x, 0)$ has a minimum at $LaTeX: x=0$. However, $LaTeX: f$ decreases with simultaneous change, $LaTeX: y=(3/2)x^2$. Along this parabola, $LaTeX: f(x,y) = -(x^4)/4$, which is negative for $LaTeX: x$ nonzero. Thus, $LaTeX: 0$ is not a local minimum of $LaTeX: f$. For further discussion about this example, see Myths and Counter Examples.