The minimum value of .y,2) = 6x – y – 6Z subject to the constraints *+y-Z = 0 and ? + 22 = 1 is?OA.O B. No correct answer.O C.7 6

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Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter10: Inequalities

Section10.8: Systems Of Linear Inequalities

Problem 2E

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The minimum value of .y,2) = 6x – y – 6Z subject to the constraints *+y-Z = 0 and ? + 22 = 1 is?
OA.
O B. No correct answer.
O C.7 6

Expert Solution

Step 1

$G i v e n, T h e m i n i m u m v a l u e o f f (x, y, z) = 6 x - y - 6 z s u b j e c t t o t h e c o n s t r a i n t s x + y - z = 0 a n d x^{2} + 2 z^{2} = 1 .$

Step 2

$N o w, f (x, y, z) = 6 x - y - 6 z - - (1) x + y - z = 0 - - (2) x^{2} + 2 z^{2} = 1 - - (3) F r o m e q u a t i o n (2), w e h a v e \Rightarrow x + y - z = 0 \Rightarrow y = z - x - - (4) F r o m e q u a t i o n (3), w e h a v e \Rightarrow x^{2} + 2 z^{2} = 1 \Rightarrow z = \sqrt{\frac{1 - x^{2}}{2}} - - (5) P u t t i n g e q u a t i o n (4) a n d (5) i n (1), w e h a v e \Rightarrow f (x) = 6 x - (\sqrt{\frac{1 - x^{2}}{2}} - x) - 6 \sqrt{\frac{1 - x^{2}}{2}} \Rightarrow f (x) = 7 x - 7 \sqrt{\frac{1 - x^{2}}{2}}$

Step 3

$∴ f (x) = 7 x - 7 \sqrt{\frac{1 - x^{2}}{2}} D i f f e r e n t i a t i n g w . r . t x, w e h a v e \Rightarrow f' (x) = \frac{d}{d x} (7 x - 7 \sqrt{\frac{1 - x^{2}}{2}}) \Rightarrow f' (x) = 7 \frac{d}{d x} (x) - \frac{7}{\sqrt{2}} \frac{d}{d x} (\sqrt{1 - x^{2}}) \Rightarrow f' (x) = 7 - \frac{7}{2 \sqrt{2} \sqrt{1 - x^{2}}} \frac{d}{d x} (1 - x^{2}) \Rightarrow f' (x) = 7 - \frac{7}{2 \sqrt{2} \sqrt{1 - x^{2}}} (- 2 x) \Rightarrow f' (x) = 7 + \frac{7 x}{\sqrt{2} \sqrt{1 - x^{2}}} F o r t h e v a l u e o f x p u t f' (x) = 0 \Rightarrow 7 + \frac{7 x}{\sqrt{2} \sqrt{1 - x^{2}}} = 0 \Rightarrow 7 \sqrt{2} \sqrt{1 - x^{2}} + 7 x = 0 \Rightarrow \sqrt{2} \sqrt{1 - x^{2}} = - x S q u a r i n g b o t h s i d e s, w e h a v e \Rightarrow 2 (1 - x^{2}) = x^{2} \Rightarrow x = \pm \sqrt{\frac{2}{3}} e i t h e r, \Rightarrow x = \sqrt{\frac{2}{3}} o r \Rightarrow x = - \sqrt{\frac{2}{3}} (N o t p o s s i b l e) T h e v a l u e o f x i s \sqrt{\frac{2}{3}} .$

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