AM-GM inequality: proof & application (Exam Question 11 of 12)

[rakeshparmar]

Sir this was absolutely helpful for JEE mains(India)

[ARKGAMING]

I would have considered LHS²-RHS² which ends up simplifying to (x-y/2)²≥0
So LHS²-RHS²≥0
LHS²≥RHS²
and since they're both positive taking the sqrt doesn't effect the direction

[akvmaths]

Class 11th maths in our country but you explained in good way 👍

[allansmith]

WHat?
Multiplying Both sides of Equation by 0 I s valid Mathematical Operation?
Please clarify me I am Confused.

[davidplanet]

(x - y)^2 = x^2 + y^2 - 2xy >=0 => x^2 + y^2 >= 2xy.
(x + y)^2 = x^2 + y^2 + 2xy >= 4xy.
Then since x > 0 and y > 0:
(x + y)/2 >= sqrt(xy) as required.

[ismahelo]

Which app or program do you use to write? By the way, great explanations!!

[particleonazock]

AM-GM inequality, Year 9 maths in Australia... Complicated proof presentation, Year maths in Australia.

[cryptocow]

Does anyone know what school Eddie Woo teaches at ?

[syuliya]

Good example, not obvious what one has to do, notice one looking at notes and not an answer to question, so one would still has to decide what to put on the exam, to be very clear that one starts in the right place for writing a proof.

[jamiewalker]

Oh gosh. Again Eddy. It is ABSOLUTELY acceptable to work from the RTP, just use if and only if arrows all the way down! Which we can do since x and y are positive. A lot of mental gymnastics over a failure to utilise reversible implication...

[xsamsungg]

I think the easier way is solving using a system of equations you get from the partial derivatives