I'm trying to find a stationary point of the function $r(u) = \gamma(\begin{matrix} \alpha -u_1+u_1^2u_2\\ \beta - u_1^2u_2 \end{matrix})$ , with $ \alpha , \beta , \gamma > 0 $
I have taken partial derivatives with respect to $u_1$ and $u_2$ and got that $$\partial_,u_1 = \gamma(\begin{matrix} -1 +2u_1u_2\\ -2u_1u_2 \end{matrix})$$
$$\partial_,u_2 = \gamma(\begin{matrix} u_1^2\\ -u_1^2 \end{matrix})$$
Therefore from the second one I came to the conclusion that $u_1$ = $0$ and that the stationary point was $u =(\begin{matrix} \alpha \gamma\\ \beta \gamma \end{matrix})$. However I'm not sure if this is correct as if $u_1=0$ then the $x$ component of $\partial_,u_1$ is still non-zero.