How to show steepest descent will not converge to local but to global minima of funtion.

Question

for a function $f(x, y) = 0.5*x^2 +0.25*y^4 −0.5*y^2$.

Show that there are infinitely many starting points for which gradient (steepest) descent will not converge to a local, let alone global, minimizer of f.

I know the global minimizer of this function is (x,y)=(0,1) and (0,-1) and (0,0) is inflection point. But I don't understand how should I show for many starting points steepest descent will not converge to a local, let alone global, minimizer of f.

Answer 1

Hint: What if you start on the line $y=0$?