Sec 14.7-sm.dvi - 728 C H A P T E R 15 DIFFERENTIATION IN...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 728 C H A P T E R 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) and, since P V = T , we get T V V P P T = - T T = - 1 Similarly, T P P V V T = V .- T V 2 . 1 P = - T V P = - T T = - 1 15.7 Optimization in Several Variables (ET Section 14.7) Preliminary Questions 1. The functions f ( x , y ) = x 2 + y 2 and g ( x , y ) = x 2- y 2 both have a critical point at ( , ) . How is the behavior of the two functions at the critical point different? SOLUTION Let f ( x , y ) = x 2 + y 2 and g ( x , y ) = x 2- y 2 . In the domain R 2 , the partial derivatives of f and g are f x = 2 x , f x x = 2 , f y = 2 y , f yy = 2 , f x y = g x = 2 x , g x x = 2 , g y = - 2 y , g yy = - 2 , g x y = Therefore, f x = f y = 0 at ( , ) and g x = g y = 0 at ( , ) . That is, the two functions have one critical point, which is the origin. Since the discriminant of f is D = 4 > 0, f x x > 0, and the discriminant of g is D = - 4 < 0, f has a local minimum (which is also a global minimum) at the origin, whereas g has a saddle point there. Moreover, since lim y g ( , y ) = - and lim x g ( x , ) = , g does not have global extrema on the plane. Similarly, f does not have a global maximum but does have a global minimum, which is f ( , ) = 0. 2. Identify the points indicated in the contour maps as local minima, maxima, saddle points, or neither (Figure 14). 1 1 1 2 3 6 10- 1- 1- 1- 2- 3- 3- 6- 10- 3 3 3 1 FIGURE 14 SOLUTION If f ( P ) is a local minimum or maximum, then the nearby level curves are closed curves encircling P . In Figure (C), f increases in all directions emanating from P and decreases in all directions emanating from Q . Hence, f has a local minimum at P and local maximum at Q . 2 6 10- 2- 6- 10 P Q In Figure (A), the level curves through the point R consist of two intersecting lines that divide the neighborhood near R into four regions. f is decreasing in some directions and increasing in other directions. Therefore, R is a saddle point. 1 1- 1- 1- 3- 3 3 3 1 R Figure (A) S E C T I O N 15.7 Optimization in Several Variables (ET Section 14.7) 729 Point S in Figure (B) is neither a local extremum nor a saddle point of f . 1 3- 1- 3 S Figure (B) 3. Let f ( x , y ) be a continuous function on a domain D in R 2 . Determine which of the following statements are true: (a) If D is closed and bounded, then f takes on a maximum value on D . (b) If D is neither closed nor bounded, then f does not take on a maximum value of D . (c) f ( x , y ) need not have a maximum value on D = { ( x , y ) : x , y 1 } . (d) A continuous function takes on neither a minimum nor a maximum value on the open quadrant { ( x , y ) : x > , y > } ....
View Full Document

This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.

Page1 / 37

Sec 14.7-sm.dvi - 728 C H A P T E R 15 DIFFERENTIATION IN...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online