If we consider the first case, we have then
$$ -4(4-2\theta_0)+\lambda=0, $$and
$$ -2(2-\theta_1)+\lambda=0. $$which yields
$$ \theta_0=\frac{16+\lambda}{8}, $$and
$$ \theta_1=\frac{4+\lambda}{2}. $$Using the constraint on \( \theta_0 \) and \( \theta_1 \) we can then find the optimal value of \( \lambda \) for the different cases. We leave this as an exercise to you.