For points $\mathbf{x} = (x, y)$ and $\mathbf{u} = (u,v)$ in $\mathbb{R^2}$, minimize $f(\mathbf{x}, \mathbf{u}) = |\mathbf{x}-\mathbf{u}|$ among all pairs of points such that $\mathbf{x}$ belongs to the plane {$\mathbf{x}\in\mathbb{R^2}:\mathbf{x}\cdot(1,2)=-10$} and $\mathbf{u}$ belongs to the parabola {$\mathbf{u}\in\mathbb{R^2}:v=u^2$}.
This question was under the section of Lagrange Multiplier. However, I am not sure if Lagrange would even be applicable to this question. Since $f:\mathbb{R^4}\to\mathbb{R}$, I am kinda lost on how I would construct a constraint function in order to include the restrictions for $\mathbf{x}$ and $\mathbf{u}$.
You have to minimize the function
$$F(x,y,u,v)=(x-u)^2+(y-v)^2$$
under the constraints
$$x+2y=-10$$
and
$$u^2-v=0.$$