<s>[INST] Inconclusive second derivative test rigorous proof I need to find and identify stationary points of the following function: 
$f(x,y) = x^4 + 2x^2y^2 - y^4 - 2x^2 + 3$
Second derivative test appears to be inconclusive (Hessian equal to zero) at the point $(0,0)$. By plotting this appears to be a maximum. How does one prove this rigorously?
 [/INST] Suppose $(x,y)$ lies in the unit disc centered at the origin (i.e. $x^2+y^2 \le 1$).
\begin{align}f(x,y) - f(0,0) &= x^4+2x^2y^2 - y^4-2x^2\\
&\le x^4 + 2x^2y^2 - y^4 - 2x^4 & x^4 \le x^2\\
&=-(x^2-y^2)^2\\
&\le 0.
\end{align}
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