Hey folks. I have some more or less qualitative questions regarding optimization problems via Lagrange multipliers. I am following the http://en.wikipedia.org/wiki/Lagrange_multipliers" [Broken] on this one and I am just a little confused by their wording.

In the first section titled "Introduction," they have the following:

1) My first question (which might seem silly) is: What is g(x,y) ? Is it a line? Or a surface? It looks like a line in their picture but I am used to functions of the forms g(x,y) being representative of surfaces; though I feel like it could be ambiguous.

2) Second, when they say: "Suppose we walk along the contour line with g = c." They don't mean that g = c IS the contour line right? They are saying that AT g(x,y) one can follow a contour line. It just seems like if g(x,y) = c is not a surface, then the whole idea of a contour is a little weird to me. So perhaps if someone can answer my first question this would make more sense to me.

Sorry if these seem stupid, but it has been awhile.

I am sketching this right now. Sketching level sets was never really my strong suit.
Is it as simple as letting c take on a certain value and then plotting that line? For example, let c = 1/2 and then plot the line y = 1/2 - 2x ? I am going to assume that the answer is yes and proceed.

And for this example, I assume that we are maximizing 'T' subject to the constraint that the point falls on the wire.

EDIT: Okay. I have plotted the circle and the line y = c - 2x for various values of c. I can see that 'T' would be maximized (and minimized) when the line y = c - 2x is tangent to the circle. I am having a couple of issues with this now.

1) Not sure how to generalize this idea beyond the simple circle constraint. That is if both my maximizing function f and my constraining function g are arbitrarily shaped surfaces, will the condition that g is 'tangent to' f be sufficient to say that f is at a maximum there?

2) I am not sure how to word this without sounding like an idiot but here goes anyway: The function T = y + 2x = c that you gave, is that a surface? That is, isn't it T(x,y) = y + 2x - c ? Or is it just T(x,y) = y + 2x ? I don't know why this is so confusing right now

T(x,y)=y+2x describes a surface. Think of T as the z coordinate. For each point in the xy plane, there's an associated temperature which you can think of as the height of the surface above the point (x,y). When you choose a specific value of T, you're looking at the intersection of that surface with the plane T=c.

T = y+2x is the temperature at a point (x,y) in the plane. I don't find it helpful to interpret it as a surface although it can be thought of that way. The level curves in this example are straight lines in the plane on which the temperature is constant.: y+2x = c. It wouldn't matter if the level curves had other shapes. Try drawing a family of hyperbolas. As long as everything is smooth the level curves will be tangent at the extremes.

For anyone who is curious, I plotted T(x,y) = y + 2x and g(x,y) = x^2 + y^2 along with the plane z(x,y) = 1. The intersection of z(x,y,) = 1 with g(x,y) = x^2 + y^2 gives the level set g(x,y) = x^2 + y^2 = 1. Here are a couple of views:

Pretty pictures, but I don't think they add to the understanding of what is going on. The example problem is given temperature T(x,y) = x+2y in the plane, find the hottest point on the wire represented by the unit circle x^{2}+y^{2} = 1. The following plot shows the level curves for T varying from -3 to 3 including, in red, T =sqrt(5) which obviously gives the max.

Maple doesn't label the contours, but in this example the y intercepts of the contour lines give T for the lines.

Different people learn in different ways I guess The picture that you drew is indeed the exact same picture that I drew originally, but I found that it did not add to my understanding of the problem.

For me the problem was that T(x,y) is in fact a surface, whether "it helps to interpret it that way" or not. And the constraint, though in this case is a simple circle, is actually a level set on a surface. So for me, being able to see that for the constraint
g(x,y) = x^2 + y^2 = anything,
T(x,y) is maximal when the level set of g(x,y) is tangent to T(x,y).