How can I show that $(-1,-1) \in \text{Argmax}\left[-\lambda_0(u_1 - 2u_2) +A(u_1-u_2)^2\right]$ implies $\lambda_0 = 0$?
The argmax is taken over $u_1$ and $u_2$ and $A$ is costant.
The source is the following: Show that the optimal control of this problem is abnormal: $$\begin{cases} \displaystyle \max \int_0^1 (u_1-2u_2)dt \\ \dot{x} = (u_1-u_2)^2 \\ x(0)=x(1) = 0 \\ |u_1| \le 1 \\ |u_2| \le 1 \end{cases} $$