<s>[INST] Find all the local extrema for $f(x,y)=x^3y^2+27xy+27y$ I have that $f'_x=3x^2y^2+27y$ and $f'_y=2x^3y+27x+27$. Setting $f'_x=0$ I get that
$$x^2=\frac{-27y}{3y^2}=-\frac{9}{y}\Leftrightarrow x=\pm\frac{3}{\sqrt{y}}i$$
Setting $x=\frac{3}{\sqrt{y}}i$ in the equation $f'_y=0$ gives
$$0=2\left(\frac{3}{\sqrt{y}}i\right)^3y+27\left(\frac{3}{\sqrt{y}}i\right)+27=\frac{27}{\sqrt{y}}i+27=0,$$ 
Which gives me the equivalent equation 
$$i+\sqrt{y}=0\Leftrightarrow \sqrt{y}=-i \Leftrightarrow y=(-i)^2=-1.$$
This is a false root however, so for $x=\frac{3}{\sqrt{y}}i$ no roots for $y$ exists. Using $x=-\frac{3}{\sqrt{y}}i$ I get instead $i-\sqrt{y}=0$ and this equation as the root $y=-1$. This means that $x=-3.$ So, one stationary point is $(x,y)=(-3,-1).$ However the book says there is another stationary point, namely $(-1,0).$ However I've found all values for $y$ and the only working value that makes sense is $y=-1$, how can I find $y=0$? It doesn't make sense for $y$ to be $0$ since that would mean division by zero when I want to get my x-value.
 [/INST] Here a way to  solve the system $\;\begin{cases}3x^2y^2+27y=0,\\[1ex]2x^3y+27x+27=0.\end{cases}$
The first equation is equivalent to $\;y(x^2y+9)=0$. So:


*

*either $y=0$, and the second equation yields $\;x=-1$;

*or $x^2y=-9$. Plugging this relation into the second equation, multiplied by $xy$ (note that $x$ has to be $\ne 0$):
$$2x^4y^2+27x^2y+27xy=0 \Rightarrow \underbrace{2\cdot 81}_{6\mkern1mu\cdot 27}-27\cdot 9+27xy=0\Leftrightarrow 27xy= 3\cdot 27\Leftrightarrow xy=3.$$
Going back to the first equation, we get $\;x^2y=3x=-9$, whence $\;x=-3 $, then $\;y=-1$.


Thus, there are two candidates: $(-1,0)$ and $(-3,-1)$.
Now calculate the hessian:
$$H_f=\begin{vmatrix}f''_{x^2}&f''_{xy}\\f''_{yx}&f''_{yx^2}\end{vmatrix}=
\begin{vmatrix}6xy^2& 6x^2y+27\\  6x^2y+27 & 2x^3\end{vmatrix}=\begin{cases}
{\scriptstyle\begin{vmatrix}0&27\\27&-2\end{vmatrix}}=-729,\\[1ex]
{\scriptstyle\begin{vmatrix}-18&-27\\-27&-2\cdot  27\end{vmatrix}}=9\cdot27\,{\scriptstyle\begin{vmatrix}2&1\\3&2\end{vmatrix}}=243.
\end{cases}$$
As the hessian is negative in the first  case, we have  a saddle point.
In the second case, it is positive, so we have a local extremum. Furthermore, the trace of the hessian is negative, which implies this extremum is a local maximum.
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