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$f(x, y) = 3x^3 − 9x + 9xy^2$

I've tried taking the partial derivatives and double derivatives of this function to solve for the local minimum, maximum values, and saddle point but haven't made much progress.

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We know that

$f_{xx}=18x$, $f_{yy}=18x$, $f_{yx}=18y=f_{xy}$.

On the other hand, we need both partial derivatives to vanish, so $9x^2-9+9y^2=0=18xy$.

This means that either $x=0$ or $y=0$. In the case that $x=0$, we also have that $y= \pm 1$, by just solving. If $y=0$, then $x= \pm 1$.

The situation is, that to determine if these points are local minima/maxima or saddle points, we should check the Hessian:

$$\begin{pmatrix}12x &18y\\18y&18x\end{pmatrix}.$$

You can either check its eigenvalues (both positive means local minimum, both negative means global maximum, and mixed means saddle point.)

However, in practice we can just take the determinant: $216x-324y=108(2x^2-3y^2)$ and plug in values. It's easy to see that $(1,0)$ and $(-1,0)$ are the max/min (the first is the min, the second is the max, you can check this by looking at values close to it) and the other two critical points $(0, \pm 1)$ are saddle points since the determinant is negative and hence the eigenvalues are mixed.

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