Solving Lagrange Multiplier Question: Find Nearest Point to Origin

[elle]

Hi, I would appreciate if anyone can help me out with the following question.

I've been asked to find the point on the surface z = xy + 1 nearest to the origin by using the Lagrange Multiplier method. But all the examples I've been given in class and for coursework gave you the constraint equation.

Is there a constraint equation given in this question?

 

[FrogPad]

Well it's not given in the sense of, constraint equation = ?
But you should be able to set one up as long as you know the distance formula.
$$d = \sqrt{(z_2-z_1)^2+(x_2-x_1)^2+(y_2-y_1)^2}$$
x, and y are going to be arbitrary and you have the expression for z. You also know that you are aiming for the orgin, so $$\vec 0 = (x_1,y_1,z_1)$$.

 

[FrogPad]

Actually 'shoot'.. that doesn't make any sense

Your constraint would be that 'the point MUST be on that surface' (that sounds more like a constraint to me).

So:
$$g(x,y,z) = z-xy=1$$
$$f(x,y,z) = d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
Where $$(x_1,y_1,z_1)=\vec 0$$

Ok, I think that makes sense now...

 

[elle]

Hmm sorry I'm still lost :(

 

[elle]

sorry I'm not sure how to use Latex, but what are the values for x2, y2 and z2? O god, I don't understand this at all :uhh:

 

[HallsofIvy]

The constraint is z = xy + 1! If you were asked simply to "find the point closest to the origin" then the answer would be (0,0) itself. But you are not asked that- you are asked to fine the point on z= xy closest to the origin.
Using the Lagrange multiplier method: The square of the distance of a point (x,y) to (0,0) (minimizing the square of distance is the same as minimizing distance itself) is minimizing x²+ y². The gradient of that is 2xi+ 2yj. The gradient of the constraint, z- xy= 1, is -yi- xj+ k. Lagrange's method says that one of those must be a multiple of the other:
2xi+ 2yj= k(-yi- xj+ k) which tells us 2x= -ky, 2y= -kx, and 0= k. What x,y,z satisfy those?

 

[elle]

hmm ok...I got x = 0, y = 0 and z = 1...is that right? so the point is ( 0 , 0 , 1 )