I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems acceptable.
I am given a set $A = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1 , x+y \ge 0 \}$ and a function $f(x,y) = (x-y)^2 (x+y)$
I am tasked with finding $f(A)$.
So because $f$ is a continuous function and the set $A$ is arch connected and bounded $f(A)$ will be an interval, more precisely $f(A) = [\min_A f, \max_A f ]$.
So now I search for the critical points of $f(A)$, I start with the interior points and construct the system
$$\begin{cases} \nabla f = 0 \\ x^2 + y^2 < 1 \\ x+y > 0 \end{cases}$$
I find that $$(x-y)(x + y) = 0 $$ so all the points of the form $x = y$ and $x = -y$ satisfy the system and I notice that the value of the function at those point is zero.
Now I want to check the boundary points so I take the set $A_1 = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 = 1 , x+y \ge 0 \}$, this is a manifold so I can use Lagrange multipliers and I obtain
$$\begin{cases} 3x^2 - 2xy - y^2 - 2 \lambda x = 0 \\ -x^2 - 2xy + 3y^2 -2 \lambda y = 0 \\ x^2 + y^2 = 1 \\ x+y > 0 \end{cases}$$
From the fist two equations I get $(x+y)( 2(x-y) - \lambda) = 0$ so either $x = - y$ or $x = (2y + \lambda)/ 2$ but in this second case I can't get an explicit solution, or a solution that works with the function I am given.
How could I solve this system in a satisfactory way?