Proof... maybe differentiation?

The sum of the squares of two positive numbers is a. Prove that their product is the maximum possible when the two numbers are equal.

So I’ve tried multiple methods but none of them work so I could really do with some help now.

I called my two numbers x and y and have the following:

x^2 + y^2=a

The sum of the squares of two positive numbers is a. Prove that their product is the maximum possible when the two numbers are equal.

So I’ve tried multiple methods but none of them work so I could really do with some help now.

I called my two numbers x and y and have the following:

x^2 + y^2=a

The sum of the squares of two positive numbers is a. Prove that their product is the maximum possible when the two numbers are equal.

So I’ve tried multiple methods but none of them work so I could really do with some help now.

I called my two numbers x and y and have the following:

x^2 + y^2=a

Have you heard of AM-GM inequality? (I heard this is on AS spec now.)

$\displaystyle \frac{a+b}{2} \geqslant \sqrt{ab}$ for non-negative $a,b$.

AM-GM inequality is definitely the quickest way to do this if you're allowed to assume it

Geometric insights which may hel[p
1) the point (x,y) lies on a circle of radius root(a)
2) the product of the two numbers is the area of the rectangle.
It's fairly easy to see that the answer is right when you draw the rectangle(s) which is defined by the origin and the point on the circle. It is maximised when its a square.

A "complex" way to do this is Lagrange multipliers, but you've probably not covered this. A similar way would be to write
x = sqrt(a - y^2)
substitute this into the product
x*y
and then differentiate wrt y and set equal to zero?

Sure differentiation is one way to go about it.

$z=xy$ and you want to maximise $z$. This is the same as maximising $Z=z^2$, where $Z = x^2y^2$ (as $x,y > 0$ anyway)

You know that $x^2+y^2=a$ so $Z = x^2(a^2-x^2)$. So... for what value of $x$ do you obtain maximum for $Z$??

You want to maximise xy, but that's a bit tricky. Would you be able to maximise $x^2 y^2$ more easily? You would have to justify the squaring, but they give you what you need in the question.

Your approach is fine. Let P=xy, use your other equation to find P as a function of x, then solve dP/dx = 0 to find x in terms of a and then y. Will be the same.

Write y implicitly as a function of x: $y = \pm \sqrt{a-x^2}$
so the function we want to maximise is
$F(x) = x\cdot y(x) = \pm x\sqrt{a-x^2}$ then try to maximise that remembering your expression for y after you differentiate it.

Have you heard of AM-GM inequality? (I heard this is on AS spec now.)

$\displaystyle \frac{a+b}{2} \geqslant \sqrt{ab}$ for non-negative $a,b$.

AM-GM inequality is definitely the quickest way to do this if you're allowed to assume it

Check out question 11 of the Edexcel AS maths sample assessment material:

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

The proof for AM-GM inequality for two terms is a two mark question - which suggests to me that strong familiarity with the result is expected. But I am neither teaching nor studying the new specification so I could be mistaken.

It's not on spec, but they can still ask for the proof of the two-term case because even if you've never seen it before, you can simply manipulate it to get (a-b)^2 >= 0

If you do that you have to use iff statements and be sure argument is reversible (and I don't expect this in AS), otherwise the logic is faulty - statement implying a truth does not make the statement true, we'd rather have a truth imply desired statement.

Easiest way to find out is to ask @Notnek. I swear this was a thing on the new spec now for Edexcel at least.

Are you asking whether the AM-GM inequality is part of the spec? If you are then the answer is no and most students won't have heard of this inequality or know what AM and GM mean. This is just a random proof that appeared in the specimen paper and probably won't come up again.

At AS students are taught one technique of proof which is to start from the result and work towards something that is true (the Edexcel textbook calls this "jottings" ). This is how I'd expect most students to start this question. But this is just a technique so if they use this they then have to then start from the fact and work towards the result.

It seems like exam boards are keen to throw in a hard proof question into their exams that a small minority of students will be able to do.

Indeed. This is quite an unusual question for AS, and I saw that other thread about the modular arithmetic question too... Seems like they're spicing things up a bit.

If you do that you have to use iff statements and be sure argument is reversible (and I don't expect this in AS), otherwise the logic is faulty - statement implying a truth does not make the statement true, we'd rather have a truth imply desired statement.

Easiest way to find out is to ask @Notnek. I swear this was a thing on the new spec now for Edexcel at least.