AS maths proof question

https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/as-l3-mathematics-sams.pdf

I'm struggling with q11 on the specimen paper shown above. I don't get how you're able to jump from the square root of x*y to root(x-y)^2. Also, why would you do this?

Reply 1

Ryanzmw

16

Ah so this actually a REALLY famous inequality called the 'Arithmetic Mean Geometric Mean' (AM-GM) inequality! For two variables the proof isn't too hard so don't worry! :biggrin:

You want to think how can you get rid of the square root? After doing that play around with the expression and try factorise! If you're having problems beyond that just reply and we'll help!

Original post by dont know it

https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/as-l3-mathematics-sams.pdf

I'm struggling with q11 on the specimen paper shown above. I don't get how you're able to jump from the square root of x*y to root(x-y)^2. Also, why would you do this?

Cf. The AM-GM inequality actually holds for n-variables i.e.

Unparseable latex formula:

$$\sqrt[n]{(a_1 \cdot a_1 \cdot \dots a_n)} \leq \dfrac{1}{n} \sum_{i = 1}^n (a_i) $$

cool, eh?

Another cool thing about your 2 variable AM-GM inequality is that it can be used to prove that a square is the shape with maximum area to perimeter ratio (it achieves equality of the AM-GM inequality)

(edited 5 years ago)

Original post by dont know it

https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/as-l3-mathematics-sams.pdf

I'm struggling with q11 on the specimen paper shown above. I don't get how you're able to jump from the square root of x*y to root(x-y)^2. Also, why would you do this?

It's unclear as to what exactly you're confused about.

I don't see where they're jumping from

\sqrt{xy}

to

\sqrt{(x-y)^2}

Reply 3

dont know it

OP

9

Original post by RDKGames

It's unclear as to what exactly you're confused about.

I don't see where they're jumping from

\sqrt{xy}

to

\sqrt{(x-y)^2}

Ah I should have mentioned it says that in the mark scheme. They give that as a way to proof it however I don't get why or how they were able to do that.

Reply 4

MR1999

21

Original post by dont know it

Ah I should have mentioned it says that in the mark scheme. They give that as a way to proof it however I don't get why or how they were able to do that.

Well if you play around with what you're given, you get

x - 2\sqrt{xy} y \geq 0.

This should look really familiar to you and so proving the inequality from there is simple.

(edited 5 years ago)

AS maths proof question

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