Yep that was my method, I thought it was a really nice use of AM-GM. I came across a nice inequalities document too on google, might link.
I must admit, the AM-GM solution would be a bit harder to find if I didn't know Lagrange multipliers. (As the appropriate definition of y_k could potentially take a little time to spot.)Yep that was my method, I thought it was a really nice use of AM-GM. I came across a nice inequalities document too on google, might link.
Awkward moment when the first proof is:
That works for AM-GM though.Awkward moment when the first proof is:
which start off with
You can do that for AM-GM?I'm talking about assuming what was to be proved.
It's fine, all steps are obviously biconditional so it doesn't matter what end you start at. And you need this biconditionality to get the "iff" statement.I'm talking about assuming what was to be proved.
This, which is why I put as obviously the squaring will still preserve the direction of the inequality.It's fine, all steps are obviously biconditional so it doesn't matter what end you start at. And you need this biconditionality to get the "iff" statement.
To add on to this, think of it as building a two-laned road all one go rather than building one lane and then once you reach one end, go backwards to build the other lane.It's fine, all steps are obviously biconditional so it doesn't matter what end you start at. And you need this biconditionality to get the "iff" statement.
This isn't too hard, will have to do though until I think of something better.
Consider a polynomial with roots for
Find the sum of the roots two at a time.
Ahh yes I see that, editing.Yep that's correct, but you double count I think, just divide by 2.
So far I have this, not sure if it'll go anywhere. Will edit in my solution as I progress.I like this sum:
Hmm I just tried a completely different method and gotI like this sum: