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Kuhn Tucker conditions

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Help to clarify matrices, Jacobian, quasi-concavity, Hessian, and Kuhn-Tucker conditions is provided.

1. (a) Solve the problem of minimizing x2 - 4x + 1 subject to x4 - 1 ≤ 0.

(b) Solve the problem of maximizing ƒ(x) = x subject to g(x) = x4 if x ≤ 0, 0 if x Є [0, 1], (x-1)4 if x ≥ 1. Explain why the Kuhn-Tucker conditions are inapplicable in this case.

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Kuhn Tucker conditions are described.

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please refer to the attachment.

1. (a) Solve the problem of minimizing x2 - 4x + 1 subject to x4 - 1 ≤ 0.

The Lagrangian Equation: (let b>0)
L= x2 - 4x + 1+ b(x4 - 1)
First Order Condition:
dL/dx=2x-4+4b x3 =0 (1)

Kuhn-Tucker condition:
b>=0, x4 - 1 ≤ 0
b( x4 - 1) = 0

if b=0, then from (1),we know x=2
but ...

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