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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 âˆ‚ T âˆ‚ V Â· âˆ‚ V âˆ‚ P Â· âˆ‚ P âˆ‚ T =  T T =  1 Similarly, âˆ‚ T âˆ‚ P Â· âˆ‚ P âˆ‚ V Â· âˆ‚ V âˆ‚ T = V Â· . T V 2 . Â· 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 â†’âˆž g ( , y ) = âˆž and lim x â†’âˆž g ( x , ) = âˆž , 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 ) : â‰¤ x , y â‰¤ 1 } . (d) A continuous function takes on neither a minimum nor a maximum value on the open quadrant { ( x , y ) : x > , y > } ....
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 Spring '08
 Hubscher
 F E R E N T, S E V E R A L VA R

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