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15.8.Ex41 - 798 C H A P T E R 15 D I F F E R E N T I AT I O...

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798 C H A P T E R 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) (c) The equality obtained in part (b) implies that λ is the rate of change in the maximum value of f ( x , y ) , subject to the constraint g ( x , y ) = c , with respect to c . 41. Let B > 0. Show that the maximum of f ( x 1 , . . . , x n ) = x 1 x 2 · · · x n subject to the constraints x 1 + · · · + x n = B and x j 0 for j = 1 , . . . , n occurs for x 1 = · · · = x n = B / n . Use this to conclude that ( a 1 a 2 · · · a n ) 1 / n a 1 + · · · + a n n for all positive numbers a 1 , . . . , a n . SOLUTION We first notice that the constraints x 1 + · · · + x n = B and x j 0 for j = 1 , . . . , n define a closed and bounded set in the n th dimensional space, hence f (continuous, as a polynomial) has extreme values on this set. The minimum value zero occurs where one of the coordinates is zero (for example, for n = 2 the constraint x 1 + x 2 = B , x 1 0, x 2 0 is a triangle in the first quadrant). We need to maximize the function f ( x 1 , . . . , x n ) = x 1 x 2 · · · x n subject to the constraints g ( x 1 , . . . , x n ) = x 1 + · · · + x n
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