Question

Show that 2|ab| <= a^2 + b^2

Answer 1

One hint I can give you is that \(|a|=\sqrt{a^2}\).

Answer 2

to get you started, 2 |ab| is just equal to 2ab when both a and b are positive, right?

and it is also equal to 2ab when both a and b are negative... because inside the absolute value function, a negative a value multiplied by a negative b value will produce a positive value overall... so it's again = 2ab

Answer 3

@wio that's a good hint too :)

Answer 4

\[\sqrt{a^2}(\sqrt{b^2}+\sqrt{b^2}) \le a^2+ b^2\]

Answer 5

i dont really know where to go from there.

Answer 6

@JakeV8 can you help at all?

Answer 7

uh... (easier to offer hints than to solve!!)

I will try...

Answer 8

jake :), i suggest to subtract both side 2ab as you stated earlier :) then a square of a number is always >= 0

Answer 9

reinforcements have arrived! That sounds like a good idea... I had a sense about it, but I wasn't seeing the next step yet.

I also noticed that the other 2 cases that I didn't mention, when either a OR b, but not both, is negative... can be rewritten on the left side of the inequality as -2ab

but I wasn't sure how to incorporate that idea...

Answer 10

yeah either way, -2ab or 2ab, you still end up with a positive number :)

Answer 11

hmmmm, ok thank you!

Answer 12

so then 0 <=a^2 + b^2 - 2ab ?

Answer 13

\[|a|^2 + |b|^2 - 2|a||b|=(|a|-|b|)^2\]

Answer 14

i should use an equality to prove an inequality?

Answer 15

How about you try factoring \(a^2-2ab+b^2\)? Can it be done?

Answer 16

(a-b)^2

Answer 17

Yes, now... regardless of whether \(a-b\) is positive or negative... if you square it will it be greater than or equal to \(0\)?

Answer 18

Just remember to keep your absolute value signs like @mukushla did.

Answer 19

Yes, it would be ..

2|a||b| <= (|a|-|b|)^2 - 2|a||b| ?

Answer 20

is that proving it?

Answer 21

No. The point is that when you factored \(a^2-2|ab|+b^2\) you ended up with \((|a|-|b|)^2\)

Answer 22

soo. (|a|-|b|)^2 >= 0 right?

Answer 23

Yes, because you can't square a number and get a negative number. Make sense?

Answer 24

yes it does!! ok, thank you!