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By Kedlaya K.S.

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ASSUMING the derivative exists, it can be computed by taking partial derivatives along a basis. 5 Convexity in several variables A function f defined on a convex subset of a vector space is said to be convex if for all x, y in the domain and t ∈ [0, 1], tf (x) + (1 − t)f (y) ≥ f (tx + (1 − t)y). Equivalently, f is convex if its restriction to any line is convex. Of course, we say f is concave if −f is convex. The analogue of the second derivative test for convexity is the Hessian criterion. A symmetric matrix M (that is, one with Mij = Mji for all i, j) is said to be positive definite if M x·x > 0 for all nonzero vectors x, or equivalently, if its eigenvalues are all real and positive.

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. Unless one can prove that the given function is convex, and thus that an interior extremum must be a global one, one must also check all boundary situations, which is far from easy to do when (as often happens) these extend to infinity in some direction.

If y is any vector and x is in the domain of f , we say the directional derivative of f along x in the direction y exists and equals fy (x) if f (x + ty) − f (x) fy (x) = lim . t→0 t If f is written as a function of variables x1 , . . , xn , we call the directional derivative along the i-th standard basis vector the partial derivative of f with respect to i and denote it by ∂f . In other words, the partial derivative is the derivative of f as a function of x i along, ∂xi regarding the other variables as constants.

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