By Kiran Kedlaya
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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.
Hence Sylvester’s criterion is indeed necessary for positive definiteness. We show the criterion is also sufficient by induction on n. BLAH. 5 1. (IMO 1968/2) Prove that for all real numbers x1 , x2 , y1 , y2 , z1 , z2 with x1 , x2 > 0 and x1 y1 > z12 , x2 y2 > z2 , the inequality 8 1 1 ≤ + 2 2 (x1 + x2 )(y1 + y2 ) − (z1 + z2 ) x1 y1 − z1 x2 y2 − z22 is satisfied, and determine when equality holds. (Yes, you really can apply the material of this section to the IMO! 6 Constrained extrema and Lagrange multipliers In the multivariable realm, a new phenomenon emerges that we did not have to consider in the one-dimensional case: sometimes we are asked to prove an inequality in the case where the variables satisfy some constraint.
2 Extrema and the first derivative The derivative can be used to detect local extrema. A point t is a local maximum (resp. minimum) for a function f if f (t) ≥ f (x) (resp. f (t) ≤ f (x)) for all x in some open interval containing t. Theorem 25. If t is a local extremum for f and f is differentiable at t, then f (t) = 0. Corollary 26 (Rolle). If f is differentiable on the interval [a, b] and f (a) = f (b) = 0, then there exists x ∈ [a, b] such that f (x) = 0. So for example, to find the extrema of a continuous function on a closed interval, it suffices to evaluate it at • all points where the derivative vanishes, • all points where the derivative is not defined, and • the endpoints of the interval, since we know the function has global minima and maxima, and each of these must occur at one of the aforementioned points.