AM-GM inequality proof help Watch
Hi, I was trying to rediscover this ingenious proof by Cauchy, but couldn't think of any clever substitution in part (b).
Things I have tried:
a=4x/3, b=4y/3, c+d=4z/3 gives cd=9/16 z(xyz)^1/3. Ugly!! Showing existence of positive solutions for c and d drove me nuts
I am pretty sure there are nicer solution.
a=(x)^4/3, b=(y)^4/3 and (cd)=z^4/3
Could anyone give me a hint or two for part b, if it is possible to give a hint without telling the answer?
I think I will be able to do part (c) once I get the part (b) right but I could be wrong.
So, you want to go from knowing
by choosing a suitable value for d.
It's hard to hint without giving away a lot but consider this:
You want to choose d as a function of a, b, c.
Because of symmetry, a symmetric function of a, b, c seems a good guess.
You know the AM-GM expression gives equality when a = b = c.
Bigger hint follows:
Consider working backwards from the 2nd expression to the first. How might you change that "^3" into a "^4"?
So, I tried to make a small hint for this, and realised that I'd actually made a massive assumption up front (that works, but counts as a fairly big hint in it's own right). So spoilering the whole thing now:
Really appreciate it!!