Transcript of page 1 of the Hotel Østerport note.

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Dear Bob,

While drinking beer yesterday several ideas came about our maximization problem and the question of root selection. You have probably solved the problem by now, but here are a few points anyhow. First, as you know, the solution

is a simple saddle point of [equation], gotten by setting both partials to zero. The problem is how to select between the 3 roots which sometimes occur. I have thought a little about why the “magic” multipliers (Lagrange) work to express constraints, and will try to briefly give you the picture: If we have a function f(x_i) to be maximized subject to constraints g_j(x_i) = c_j,[?] then we can picture the constraints as limiting our region of interest to a “surface” (subspace) within the x_i space. [erasure] There is in fact a whole family of surfaces which can be labeled by the value of the g_j(x_i) which are constant over any one surface.

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If we now choose some fixed values [lambda]₁, [lambda]₂ [lambda]_n

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