I am trying to find the farthest point from the origin to a point on the circle $$(x-2)^2+y^2=1$$ I am not great with the formatting on here but this is what I have so far...
$$f(x,y)=x^2+y^2 $$ $$g(x,y)=(x-2)^2+y^2-1$$ $$\nabla f =\lambda\nabla g$$ $$<2x,2y>=\lambda<2(x-2),2y>$$ (1)$$2x=2\lambda(x-2)$$ (2)$$2y=\lambda2y$$ From (2)$$\lambda=1$$ substitute $\lambda$ into (1) and produce 0=-4 which tells me no solution... What did I do wrong???
You have proved that $\lambda=1$ was impossible.
So from (2): $$ 0 = (\lambda-1)y \implies y = 0; \\ x - 2 = \pm 1 $$ which gives the maximizer and minimizer.