Show $(0,0)$ is a saddle for $x^6 - x^4 + y^6$

Question

The determinant of the Hessian is $0$, so we can't use this method but that's the only method I can see from the textbook. How else can we show that this is a saddle point?

Answer 1

Along the line $x=0$, the function is concave up, looking like $y^6$. Along the line $y=0$, the function is $x^6-x^4=x^4(x^2-1)$, which near 0 looks like $-x^4$, which is concave down.