Two critical points and their nature

Critical points. $f_y=0\Rightarrow y=1$. $4-2x-6x^2=0\Rightarrow 3x^2+x-2=0\Rightarrow x=-1$ or $x=2/3$. Points: $A=(-1,1)$, $B=(2/3,1)$.

Second derivatives. $f_{xx}=-2-12x$, $f_{xy}=0$, $f_{yy}=-2$.

At $A$. $H=\begin{pmatrix}10&0\\0&-2\end{pmatrix}$, $\det=-20<0$: saddle.

At $B$. $H=\begin{pmatrix}-10&0\\0&-2\end{pmatrix}$, $\det=20>0$ and $f_{xx}<0$: local maximum. $f(B)=71/27$.

Conclusion. Saddle at $(-1,1)$, local maximum at $(2/3,1)$ with value $71/27$.