AM-GM Inequality Explanation

This video aims to enlighten students interested in competition maths/olympiads on this very useful inequality. The video covers a couple of different ways to prove the inequality along with a simple example.

[madhadamard]

Hi tudy, apologies for this mistake, went completely over my head - good catch! I intended Attempt 2 to look something like: after adding 4ab to both sides to obtain a^2 + 2ab + b^2 >= 4ab you could factorise the left hand side of the inequality and then divide both sides by 4 to get ((a+b)^2)/4 => ab and then square root both sides to get (a+b)/2 >= ab^(1/2), like you pointed out. Sorry once again and thanks for pointing it out!

[tudy]

Good job. Simple but effective presentation. However, at 5:16 your algebraic proof does not work out in the way you presented it: The red flag pops up, when you add 4ab just to subtract 2ab in the immediate next step. Indeed the third line on the right is exactly the same as the second on the left side (which is the failed attempt). You then say to take roots of the third line, which would result in the exact same problem as before.

Instead your third line on the right should read: (a+b)² >= 4ab
Then you can easily take the root and get to your last line.

[linguamathematica]

Very nice! the geometric proof is quite intuitive and powerful