Explanation Procedure used: Linear independence theorem. A set of functions is said to be independent if their Wronskian is nonzero. If the functions are f 1 and f 2 . Thus, W ( f 1 , f 2 ) = | f 1 f 2 f ′ 1 f ′ 2 | W ( f 1 , f 2 ) ≠ 0 Calculation: Consider the two functions f 1 ( x ) = x 2 and f 2 ( x ) = x | x | . Since, the function f 2 ( x ) = x | x | has two definitions. On the interval ( − ∞ , 0 ) , f 2 = − x 2 and for the interval ( ∞ , 0 ) , f 2 = x 2 . Thus, the Wronskian on the interval ( − ∞ , 0 )

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3, Problem 9CR

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To find: An interval over which the two set of two functions f1(x)=x2 and f2(x)=x|x| is linearly independent. Then give an interval over which the set consisting of f1 and f2 is linearly dependent.

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Chapter 3, Problem 8CR

Chapter 3, Problem 10CR

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Here is the optimal tableau for a standard Max problem. zx1 x2 x3 24 81 82 83 rhs 1 0 5 3 0 6 0 1 .3 7.5 0 - .1 .2 0 0 28 360 0 -8 522 0 2700 0 6 12 1 60 0 0 -1/15-3 1 1/15 -1/10 0 2 Using that the dual solution y = CBy B-1 and finding B = (B-¹)-¹ we find the original CBV and rhs b. The allowable increase for b₂ is If b₂ is increased by 3 then, using Dual Theorem, the new value for * is If c₂ is increased by 10, then the new value for optimal > is i.e. if no change to BV, then just a change to profit on selling product 2. The original coefficients c₁ = =☐ a and c4 = 5 If c4 is changed to 512, then (first adjusting other columns of row0 by adding Delta times row belonging to x4 or using B-matrix method to update row0) the new optimal value, after doing more simplex algorithm, for > is

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An interval over which the two set of two functions f 1 ( x ) = x 2 and f 2 ( x ) = x | x | is linearly independent. Then give an interval over which the set consisting of f 1 and f 2 is linearly dependent.

Solution Summary: The author explains the linear independence theorem, where a set of functions is said to be independent if their Wronskian is nonzero.

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To find: An interval over which the two set of two functions f1(x)=x2 and f2(x)=x|x| is linearly independent. Then give an interval over which the set consisting of f1 and f2 is linearly dependent.

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