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15.7%20Ex%2023%20-%2033 - 744 C H A P T E R 15 D I F F E R...

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744 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) 23. Use a computer algebra system to fnd numerical approximations to the critical points oF f ( x , y ) = ( 1 x + x 2 ) e y 2 + ( 1 y + y 2 ) e x 2 Use ±igure 18 to determine whether they correspond to local minima or maxima. z x y FIGURE 18 Plot oF the Function f ( x , y ) = ( 1 x + x 2 ) e y 2 + ( 1 y + y 2 ) e x 2 . SOLUTION The critical points are the solutions oF f x ( x , y ) = 0and f y ( x , y ) = 0. We compute the partial derivatives: f x ( x , y ) = ( 1 + 2 x ) e y 2 + ³ 1 y + y 2 ´ e x 2 · 2 x f y ( x , y ) = ³ 1 x + x 2 ´ e y 2 · 2 y + ( 1 + 2 y ) e x 2 Hence, the critical points are the solutions oF the Following equations: ( 2 x 1 ) e y 2 + 2 x ³ 1 y + y 2 ´ e x 2 = 0 ( 2 y 1 ) e x 2 + 2 y ³ 1 x + x 2 ´ e y 2 = 0 Using a CAS we obtain the Following solution: x = y = 0 . 27788, which From the fgure is a local minimum. 24. Use the contour map in ±igure 19 to determine whether the critical points A , B , C , D are local minima, maxima, or saddle points. 1 1 0 0 2 3 1 1 2 3 3 2 1 0 3 2 1 3 1 02 3 1 2 A C D B FIGURE 19 The nearby level curves at A and C are closed curves encircling A and C .Aswemovetowa rds A the Function increases in all directions, while moving towards C the Function decreases in all directions. We conclude that the Function has a local maximum at A and a local minimum at C . The level curves through B and D consist oF two curves intersecting at these points respectively. These curves divide the neighborhoods near B and D into Four regions. In some oF the regions the Function is increasing and in others it is decreasing as we move towards B or D . This implies that B and D are saddle points. 25. Which oF the Following domains are closed and which are bounded? (a) { ( x , y ) R 2 : x 2 + y 2 1 } (b) { ( x , y ) R 2 : x 2 + y 2 < 1 } (c) { ( x , y ) R 2 : x 0 } (d) { ( x , y ) R 2 : x > 0 , y > 0 } (e) { ( x , y ) R 2 : 1 x 4 , 5 y 10 } (f) { ( x , y ) R 2 : x > 0 , x 2 + y 2 10 } (a) { ( x , y ) R 2 : x 2 + y 2 1 } : This domain is bounded since it is contained, For instance, in the disk x 2 + y 2 < 2. The domain is also closed since it contains all oF its boundary points, which are the points on the unit circle x 2 + y 2 = 1.
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SECTION 15.7 Optimization in Several Variables (ET Section 14.7) 745 (b) { ( x , y ) R 2 : x 2 + y 2 < 1 } : The domain is contained in the disk x 2 + y 2 < 1, hence it is bounded. It is not closed
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15.7%20Ex%2023%20-%2033 - 744 C H A P T E R 15 D I F F E R...

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