Answered: The given function subject to the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.3: Systems Of Inequalities

Problem 15E

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Find the values of x, y, z that minmize x²+y²+z²-2x-2y-2z+3 subject to the constraint 38−x−3y−z=0.

The given function subject to the constraint 38−x−3y−z=0 has a minimum at x=___________, y=__________, and z=____________.

Expert Solution

Step 1

$L e t f (x, y, z) = x^{2} + y^{2} + z^{2} - 2 x - 2 y - 2 z + 3 g (x, y, z) = x + 3 y + z = 38 \nabla f (x, y, z) = λ \nabla g (x, y, z) (f_{x}, f_{y}, f_{z}) = λ (g_{x}, g_{y}, g_{z}) c o m p u t e p a r t i a l d e r i v a t i v e o f f a n d g f_{x} = 2 x - 2 f_{y} = 2 y - 2 f_{z} = 2 z - 2 g_{x} = 1, g_{y} = 3, g_{z} = 1 f_{x} = λ g_{x} f_{y} = λ g_{y} f_{z} = λ g_{z}$

Step 2

$\Rightarrow 2 x - 2 = λ . . . . . . . . (i) 2 y - 2 = 3 λ . . . . . . . . (i i) 2 z - 2 = λ . . . . . . . . (i i i) a l s o, w e h a v e x + 8 y + z = 38 . . . . . . . . . . . (i v) s u b s t i t u t e v a l u e o f x, y a n d z f r o m e q u a t i o n (i), (i i), (i i i) i n t o e q u a t i o n (i v), w e g e t \frac{λ + 2}{2} + 3 (\frac{3 λ + 2}{2}) + \frac{λ + 2}{2} = 38 11 λ + 10 = 38 \times 2 11 λ = 66 λ = 6$

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