15.7.Ex23-25

15.7.Ex23-25 - 744 C H A P T E R 15 D I F F E R E N T I AT...

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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.
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15.7.Ex23-25 - 744 C H A P T E R 15 D I F F E R E N T I AT...

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