In the attached inequality, constrained optimisation problem solvable through KKT conditions. I have already wrote the lagrangian and the KKT conditions at the top. It ShouLld say: Min L

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x

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y

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$$lambda

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$=$

$$xy

$-$

$$lambda

$1$

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$1$

$4$

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$$Where the L stands for lagrangian. Looking at the specific case where lambda

$1$

$=$

$0$

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i

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e

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$$slack constraint

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$$and lambda

$2$

$>$

$0$

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binding constraint

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$$you can see that I have managed to find half of the critical points, however, the mark scheme identifies additional critical points from this case

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$-$

$2$

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$0$

$)$

$$and

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$2$

$,$

$0$

$)$

$$these can be seen fairly easily from inspection of the constraint. However, I am concerned that there should be a systematic way of finding them apart from just guessing values in the constraint and I think I would have missed them. The Mark scheme provides know other aditional information regarding how to solve this specific case. I do not need you to solve the other three cases of the constraints being binding and or slack

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however if you h have time that is appreciated

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$1$

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$$Is my approach in the attached correct so far?

$2$

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$$How would I have found the aditional critical points

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$-$

$2$

$,$

$0$

$)$

$$and

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$2$

$,$

$0$

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$$systematically other than just looking at the binding constraint x

$^$

$2$

$+$

$4$

y

$^$

$2$

$=$

$4$

$$and guessing?

$3$

$.$

$$Is there a better or faster way of solving for the two critical points i did find systematically points?

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