minimise [$]x + y[$] subject to [$]x^2 + y^2 - 2 = 0[$]
[$]x^2 + y^2 - 2 = 0[$] is circle radius 2
[$]x=\sqrt{2}\cos\theta[$] and [$]y=\sqrt{2}\sin\theta[$]
[$]x+y=\sqrt{2}(\cos\theta+\sin\theta)=2\sin(\theta+\pi/4)[$]
answer is -2
I agree. Nice approach. At which point does it reach a minimum based on this analysis?
Another approach is to use Lagrange multipliers [$]L(x,y,\lambda) = x + y - \lambda(x^2 + y^2 -2)[$]
Taking the gradient results in 4 solutions [$]x^2 = 1, y^2= 1, \lambda = +1/2, \lambda = -1/2[$], one of which (-1.-1) is the minimum.
Can your approach be applied to the additional inequality constraint [$] y \geq 0[$]?