Part B/C - Algebraic Inequalities (1 Viewer)

[mmmmmmmmaaaaaaa]

I completed part a
a) a^4+b^4 ≥ 2a^2b^2 etc.
for part b though, I'm unsure what it would be following the same logic

 

[Trebla]

From the result in part (a)
- Replace a with
- Replace b with
- Replace c with

 

[Run hard@thehsc]

For part b, not sure, but you could do a = b = c and then sub it in - Could someone verify!

 

[5uckerberg]

For part b, not sure, but you could do a = b = c and then sub it in - Could someone verify!

No, because if you do that you will have both sides being equal which is no longer an inequality.

 

[5uckerberg]

I completed part a
a) a^4+b^4 ≥ 2a^2b^2 etc.
for part b though, I'm unsure what it would be following the same logic
View attachment 34637

For part b I have an antidote. Do you notice how part b started with "Hence ... "
So using the same idea as part a you will see that




Expanding they become




Add them all together it becomes
Divide by 2 we will have the required answer for part b.

@Run hard@thehsc & @mmmmmmmmaaaaaaa here is your solution to part b

When you see the word "Hence ... " you tend to follow the same logic because hence implies that you are using one statement to prove the other. Works just like "Similiarly ... " but for the question. Reasoning being similiarly, is used when you do not want to write the same statement twice.

 

[5uckerberg]

I have made a small mistake in part c from my previous post but instead part c can be done very easily because of one reason. Since then we can exploit the fact that . Well that is simply .

Once that is out of the way we can now combine parts a and b and formulate an inequality. As shown it becomes
. This is simply just

So to finish it off


This response correct.

 

[mmmmmmmmaaaaaaa]

Part c will be a walk in the inequalities forest. Are you ready for that? So the first step will be to start with and .

Next, we will have to start with the fact that .
There we divide by 2 and using the inequalities that are known we will have
Noting that

At this momwnt our goal will be to multiply by 2 and then multiply by 2 and in doing so we actually multiply by 4. There you will have
.

Now the question starts.

Do you recall that
Well, on the LHS of the inequality it will become
There, .
I imagine what happens is that out of the rabbit's hat and in doing so it will give us . I assume that out of nowhere. I could sense that either this is a typo or something shady is happening. If it is the latter please show us.
Thus, we have proven that

Thanks!