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Math 2374: Multivariable Calculus and Vector Analysis Math 2374: Multivariable Calculus and Vector Analysis Part 25 Fall 2012 Extreme points Definition If f : U ⊂ Rn ! R is a scalar function, • a point x0 2 U is called a local minimum of f if there is a neighborhood V of x0 such that for all points x 2 V, f (x) ≥ f (x0), • a point x0 2 U is called a local maximum of f if there is a neighborhood V of x0 such that for all points x 2 V, f (x) ≤ f (x0), • a point x0 2 U is called a local, or relative, extremum of f if it is either a local minimum or a local maximum, • a point x0 2 U is called a critical point of f if either f is not differentiable at x0 or if rf (x0) = Df (x0) = 0, • a critical point that is not a local extremum is called a saddle point. First Derivative Test for Local Extremum Theorem If U ⊂ Rn is open, the function f : U ⊂ Rn ! R is differentiable, and x0 2 U is a local extremum, then Df (x0) = 0; that is, x0 is a critical point. Proof. Suppose that f achieves a local maximum at x0, then for all n h 2 R , the function g(t) = f (x0 + th) has a local maximum at t = 0. Thus from one-variable calculus g0(0) = 0. By chain rule 0 g (0) = [Df (x0)] h = 0 8h So Df (x0) = 0. Same proof for a local minimum. Examples Ex-1 Find the maxima and minima of the function f (x; y) = x2 + y 2. Ex-2 Consider the function f (x; y) = x2 − y 2, show that it has one critical point. Is this critical point a local extremum? Ex-3 Find all critical points of f (x; y) = x2y + y 2x. Ex-4 Find all critical points of f (x; y) = 2(x2 + y 2)e−x2−y 2 . Hessian of a scalar function Definition n 2 Suppose that f : U ⊂ R ! R is of class C at x0 2 U. The Hessian of f at x0 is a quadratic function defined by n 1 X @2f Hf (x0)(h) = (x0)hi hj 2 @xi @xj i;j=1 0 2 2 1 @ f ··· @ f 0 h 1 @x1@x1 @x1@xn 1 1 B . C . = (h1;:::; hn) B . C B . C 2 @ A @ A @2f @2f ··· hn @xn@x1 @xn@xn | {z } M2(x0) Remark By equality of mixed partial derivative, the second derivative matrix M2(x0) is symmetric. Hessian and second-order Taylor formula n 2 Suppose that f : U ⊂ R ! R is of class C at x0 2 U and suppose that x0 is a critical point of f ; that is, Df (x0) = 0. Then the second-order Taylor formula reduces to: f (x0 + h) = f (x0) + Hf (x0)(h) + R2(x0; h): ) at a critical point the Hessian equals the first non constant term in the Taylor series of f . Idea: study the Hessian to determine if a critical point is a local extremum of f . Second Derivative Test for Local Extremum Definition A quadratic function g : Rn ! R is called positive-definite if g(h) ≥ 0 for all h 2 Rn and g(h) = 0 only for h = 0. Similarly, g : Rn ! R is called negative-definite if g(h) ≤ 0 for all h 2 Rn and g(h) = 0 only for h = 0. Theorem n 3 Suppose that f : U ⊂ R ! R is of class C at x0 2 U and that x0 is a critical point of f . • If the Hessian Hf (x0) is positive-definite, then x0 is a relative minimum of f . • If the Hessian Hf (x0) is negative-definite, then x0 is a relative maximum of f . 2 2 2 2 Example: f (x; y) = x + y and Hf (0)(h) = h1 + h2. Determinant Test for Posi/Nega-tive Definiteness Theorem Let f (x; y) be of class C3 on an open set U in R2. A point (x0; y0) is a (strict) local minimum of f provided the following three conditions hold: @f @f (i) (x ; y ) = (x ; y ) = 0 @x 0 0 @y 0 0 @2f (ii) (x ; y ) > 0 @x2 0 0 2 @2f @2f @2f (iii) D = − > 0 at (x ; y ) @x2 @y 2 @x@y 0 0 (D is called the discriminant of the Hessian.) If in (ii) we have < 0 instead of > 0 and condition (iii) is unchanged then we have a (strict) local maximum. Remark If D < 0, then we have a saddle-point. If D = 0, we say that the critical point is degenerate. Examples Ex-1 Classify the critical points of the function f (x; y) = x2 − 2xy + 2y 2. Ex-2 Locate the relative maxima, minima and saddle-points of the function f (x; y) = ln(x2 + y 2 + 1) Ex-3 Same question for the function f (x; y) = 1=xy. Ex-4 Analyze the behavior of f (x; y) = x5y + xy 5 + xy at its critical point. Global Maxima and Minima Definition n Suppose that f : A ⊂ R ! R. A point x0 2 A is said to be an absolute maximum ( or absolute minimum ) point for f if f (x) ≤ f (x0) ( or f (x) ≥ f (x0)) for all x 2 A. Definition A set D ⊂ Rn is said to be bounded if there is a number M > 0 such that kxk < M for all x 2 D. A set is said to be closed if it contains all its boundary points. Theorem (Global Existence for Maxima and Minima) Let D be a closed and bounded set of Rn and let f : D ! R be continuous. Then f assumes its absolute maximum and minimum values at some points x0 and x1 of D. Strategy for finding Maxima and Minima Let f be continuous functions of two variables defined on a closed and bounded region D of the plane, which is bounded by a smooth closed curve (D = U [ @U). To find the absolute maximum and minimum of f on D: (i) Locate all maxima and minima for f in U. (ii) Find the maximum and minimum of f viewed as a function only on @U. (iii) Compute the values of f at all these critical points. (iv) Compare all these values and select the largest and the smallest. Example: Find the maximum and minimum values of the function f (x; y) = x2 + y 2 − x − y − 1 in the unit disk D..
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