By Kedlaya K.S.
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Littlewood, and G. P´olya, Inequalities (second edition), Cambridge University Press, Cambridge, 1951.  T. Popoviciu, Asupra mediilor aritmetice si medie geometrice (Romanian), Gaz. Mat. Bucharest 40 (1934), 155-160.  S. Rabinowitz, Index to Mathematical Problems 1980-1984, Mathpro Press, Westford (Massachusetts), 1992.
3 Convexity and the second derivative As noted earlier, a twice differentiable function is convex if and only if its second derivative is nonnegative. 4 More than one variable Warning! The remainder of this chapter is somewhat rougher going than what came before, in part because we need to start using the language and ideas of linear algebra. Rest assured that none of the following material is referenced anywhere else in the notes. We have a pretty good understanding now of the relationship between extremization and the derivative, for functions in a single variable.
31 The Lagrange multiplier criterion for an interior local extremum of the function f (x 1 , . . , xn ) under the constraint g(x1 , . . , xn ) = c is the existence of λ such that ∂f ∂g (x1 , . . , xn ) = λ (x1 , . . , xn ). ∂xi ∂xi Putting these conditions together with the constraint on g, one may be able to solve and thus put restrictions on the locations of the extrema. ) It is even more critical here than in the one-variable case that the Lagrange multiplier condition is a necessary one only for an interior extremum.