I have confusion regarding Lagrange's multiplier. I was referring to this wiki article http://en.wikipedia.org/wiki/Lagrange_multiplier.
It says that if I have the two contours of the original function $f(x,y)$ to be minimized and the constraint $g(x,y)$ then the gradient of $f(x,y)$ and gradient of $g(x,y)$ are parallel when the contour lines of $f$ and $g$ touch and the tangent vectors of the contour lines are parallel.
And it gives the condition
$\nabla_{x,y}f = -\lambda\cdot \nabla_{x,y}g$
where did this condition come from? I mean they have placed $\lambda$ to make the values equal. But how come we have the negative sign? I didn't get this equation.
Also how did this condition lead to the following
$\mathcal L(x,y,\lambda) = f(x,y) + \lambda \cdot g(x,y)$
Also I have one more question, the wiki article says that we can choose the value of $x$,$y$ such that the contour of $g$ and $f$ touch each other tangentially. They have given an example in the figure at the top right showing it, for maximization. But I am not sure why they need to touch tangentially what if they cross each other can we take that point. For example consider the same figure lets say I want to minimize the function with the constraints. If I take the point when the two contours intersect which is for value $d_2$ as given for the wikipedia figure, definitely I can take this as $x$,$y$ even though they don't touch tangentially. I am a bit confused. So can anyone please explain?