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1 solution
Nice, now here is a bonus question. There is a small error in the following approach: 5 x + 2 x + y + y + 2 y ≥ ( 4 x 2 y 3 ) 0 . 2 5 3 x + 4 y ≥ ( 2 4 ) 0 . 2 3 x + 4 y ≥ 9 . 4 4 So, where is the mistake :) ?
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I'm explaining this poorly but is it because in that approach you aren't choosing the 5 terms in the arithmetic mean such that they are equal, as in x is not equal to 2x and y is not equal to 2y? I'm trying to familiarize myself with AM>=GM so you've posed an interesting question for me to ponder lol. In my understanding you have to check that the terms can indeed be equal so that the equality for the lower bound exists (or something like that).
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Correct, if we want to find the minimum value using the AM-GM inequality, then we have to make sure that equality can exist among the taken variables, which can surely not take place in the above mentioned case. Thus, basically the error in the above method is that we took ≥ sign. Therefore 3 x + 4 y ≥ 9 . 4 4 is actually incorrect while 3 x + 4 y > 9 . 4 4 is absolutely correct.
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@Abhijeet Verma – Thanks for the extra question! And that's an interesting point that 3x+4y>9.44 is still true. I wouldn't have thought about removing the equality part without you having brought it to my attention. It seems like that could be useful if you're not needing the highest lower bound or lowest upper bound or whichever case you're working with and you just want a semi-decent estimate using AM>=GM.
i think you shall mention that x and y are greater that 0 because if they are not then AM-GM cannot be applied