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
gradient(f) = -lambda* gradient(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
L(x,y,lambda) = f(x,y) + lambda * 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 d2 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?