= Solution
The <stationary point> equations give $y=x$ and $x^3=x$, hence $(0,0),(1,1),(-1,-1)$. At $(0,0)$ the <Hessian matrix> has negative <determinant> $-16$, so \b[$(0,0)$ is a <saddle point>]. At each of $(1,1)$ and $(-1,-1)$, the <Hessian matrix> has positive leading minors $12$ and $32$, proving a strict <local minimum>. More strongly,
$$
f(x,y)=(x^2-1)^2+2(y-x)^2+3\ge3.
$$
Thus \b[$(1,1)$ and $(-1,-1)$ are exactly the <global minima>, with value $3$]. There are no <local maxima>, since any differentiable interior <local maximum> would be a <stationary point>, and all three have been classified.
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