Question

# Consider the following optimization problem:

Max x_{1}^{2}+

x_{2}^{2}

subject to : 5x_{1}^{2}+ 6x_{1}x_{2} + 5x_{2}^{2} ≤ 1

where x_{1} and x_{2} are choice variables:

1) Write the Lagrangean and the Kuhn-Tucker conditions.

2)State and verify the second order condition. Distinguish between sufficient and necessary condi-

tions.

3)Is the constraint qualification condition satisfied? Show clearly why or why not.

4) Solve the Kuhn-Tucker conditions for the optimal choice: x_{1}, x_{2} and λ, Use the second order

condition to analyze your solution and determine which of your stationary points is a maximum.

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