Important definitions related to maximum and minimum.

20. Maxima and minima: IRecall that K ⊂ Rn is closed if the complement is open. Recall also that this is equivalent to saying that K contains all of its limit points. Definition 20.1. We that K ⊂ Rn is bounded if there is a real numbbe M such that �x�≤ M, for all x ∈ K. We say that K is compact if K is closed and bounded. Note that K is bounded if and only if K ⊂ BM (O), where O is the origin. Example 20.2. (1) [a,b] ⊂ R is compact. (2) (a,b] ⊂ R is bounded but not closed (whence not compact). (3) [a, ∞) ⊂ R is closed but not bounded (whence not compact). (4)K = { x ∈ Rn |�x�≤ M },is compact. (5)K = { x ∈ Rn |�x� 0, then a is a local minimum of f. (iii) If f�(a)=0 and f��(a) = 0, then we don’t know. The reason why the second derivative works follows from Taylor’s Theorem with remainder, applied to the second Taylor polynomial. To figure out the multi-variable form of the second derivative test, we need to consider the multi-variable second Taylor polynomial: P�a,2f(�x)= f(�a)+ �f(�a) �h +1�htHf(�a)�h.· 2 Recall that �h = (h1, h2, . . . , hn) and Hf(�a) = ( ∂2f ∂xi∂xj (�a)). The important term is then Q(�h)= htHf(�a)h. 2 �� � � � � Definition 20.9. If A is a symmetric n × n matrix, then the functionQ(�h)= �x tA�x, is called a symmetric quadratic form. We say that Q is positive definite if �x =0 implies that Q(�x) > 0. We say that Q is positive definite if �x =0 implies that Q(�x) < 0. Example 20.10. If A = I2 then � �� � 10 x 22Q(x,y)=(x,y)= x + y,01 y which is positive definite. If A = −I2 then Q(x,y)= −x2 − y2 is negative definite. Finally if 10 0 −1 then Q(x,y)= x2 − y2 is neither positive nor negative definite. Proposition 20.11. If �a ∈ K ⊂ Rn is an interior point and f : K −→ R is C3 and �a is a critical point, then (1) If Q(�h)= �htHf(�a)�h is positive definite, then �a is a minimum. (2) If Q(�h)= �htHf(�a)�h is negative definite, then �a is a maximum. (3) If Q(�h)= �htHf(�a)�h is not zero and is neither positive nor negative definite, then �a is a saddle point. Proof. Immediate from Taylor’s Theorem. � Proposition 20.12. If A is a n × n matrix, then let di be the determinnan of the upper left i × i submatrix. Let Q(�h)= htAh. (1) If di > 0 for all i, then Q is positive definite. (2) If di > 0 for i even and di < 0 for i odd, then Q is negative definite. Let’s consider the 2 × 2 case. ab A = bd In this case Q(x,y)= ax 2 +2bxy + cy 2 . Assume that d1 = a> 0. Let’s complete the square. a = α2, some α> 0. Q(x,y)=(αx + b/αy)2 +(d − b2/α2)y 2 =(αx + b/αy)2 +(ad − b2)/ay2 . In this case d1 = a> 0 and d2 = ad − b2 . So the coefficient of y2 is positive if d2 > 0 and negative if d2 < 0. 3 MIT OpenCourseWarehttp://ocw.mit.edu 18.022 Calculus of Several Variables Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.