What is the approach to solve simple constrained optimization when first order condition $\nabla f = 0$ yields solution outside of domain

Question

I wish to solve the problem

min $ f(x_1, x_2) = x_1^2 - x_1 + x_2 + x_1x_2$

subj $x_1\ge 0, x_2 \ge 0$

We find $\nabla f = [2x_1 - 1 + x_2, 1+x_1] = 0$ yields $x_1 = -1, x_2 = 3$ which is outside the domain

What are some approaches to proceed from here to find the minimum which is $x_1 = 1/2, x_2 = 0$

Answer 1

Basically you've determined that the derivative doesn't help, now you have to check the boundaries of the domain, that is when $x_1 = 0$ or when $x_2 = 0$, the first yields $$f(0 , x_2) = x_2$$ the minimum value of this on the domain $x_2 \geq 0$ is $x_2 = 0$. The second condition gets us $$f(x_1, 0) = x_1^2 - x_1$$ Differentiating that yields $$f'(x_1, 0) = 2x_1 - 1$$ Which shows a critical point at $x_1 = \frac{1}{2}$, thus our extreme value is at $(x_1, x_2) = \left(\frac{1}{2},0 \right)$