#max of sum

269 messages · Page 1 of 1 (latest)

sudden hare
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i didn't solve taht someone can explain a method not to solve this thx

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zenith relicBOT
craggy sapphire
sacred kettle
craggy sapphire
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There’s 2 main approaches here

craggy sapphire
craggy sapphire
sudden hare
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sould i remplace x by sqrt( 1 -y square) ?

craggy sapphire
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And the other is parametrisation

sacred kettle
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tbfair on second thought that sounds really tedious (wrt substitution)

craggy sapphire
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First off

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You can simplify your expression

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Into a simpler problem

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Before doing anything else

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Do you see how? @sudden hare

sudden hare
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mmmh

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lemme take a moment

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remplace x by cos(x) ? and y by sin(x) ?

craggy sapphire
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That’s one approach yes

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But don’t do it yet

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You can simplify the problem

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Just by doing some algebraic manipulation

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Do you see how?

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Also there is also a neat secret third approach, using AM-GM

sudden hare
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AM-GM ?

craggy sapphire
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It’s an inequality

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But forgot that for a moment

sudden hare
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okok

craggy sapphire
sudden hare
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nop

craggy sapphire
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Okay, is it okay if I walk you through that?

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Or do you want a hint?

sudden hare
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yes yes

sudden hare
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i need the method npt the solution

craggy sapphire
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Yes

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So this is only a first step

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To make the problem somewhat easier

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Try and make use of the fact that x^2 + y^2 = 1

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In the expression -x^2 + 10xy + 23y^2

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And simplify it, by adding 0

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That’s the hint

sudden hare
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Should I substitute?

craggy sapphire
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If possible yes

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But you need to add 0 first

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In a certain way

sudden hare
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add 0 ? where

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there is a web with math writing ?

craggy sapphire
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You can use latex here

craggy sapphire
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As terms

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But that’s far from x^2 + y^2

sudden hare
craggy sapphire
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To make it look like x^2 + y^2

sudden hare
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oh ok $

craggy sapphire
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Up to some factor

sudden hare
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not need compex number ?

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complex

craggy sapphire
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No

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Let me maybe give you an example

sudden hare
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yes pls

craggy sapphire
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Say we knew that ab=1, and we wanted to simplify a^2b

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What could we do?

sudden hare
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b =a-1 so 2b =2(a-1) so a^b=a^a(a-1)

craggy sapphire
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Okay let me try something else

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Say we knew a = 1

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And we wanted to simplify a+b

sudden hare
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and ab=1 ?

craggy sapphire
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No ignore that

sudden hare
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okok

craggy sapphire
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New example

sudden hare
craggy sapphire
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Mhm

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Now, in our problem above

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We can think of x^2 + y^2 as being the same as a

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I.e a = 1

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And we want to find a in -x^2 + 10xy + 23y^2

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Now how do we do this?

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We something very close it

sudden hare
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ohhhh

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i understand

craggy sapphire
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-x^2 + 23y^2

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Is almost the same as x^2+y^2

sudden hare
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yes

craggy sapphire
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But we need to add 0

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To change perspective

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In a cleaver way

sudden hare
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ok lemme take a moment

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i don't find

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I don't see where you're going with this.

craggy sapphire
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Want me to reveal it?

sudden hare
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yeah

craggy sapphire
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add 23x^2 - 23x^2 to - x^2 + 10xy + 23y^2

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I’m adding 0 here

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So nothing changes

sudden hare
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AHHHHH

craggy sapphire
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Mhm

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What happens after simplifying?

sudden hare
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-x^2 + 10xy +23

craggy sapphire
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Almost

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Don’t forget the -23x^2

sudden hare
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-23x^2

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wait I find somrthing

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so

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if x^2+y^2=1 x=1 or -1 and same for y

craggy sapphire
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Hm?

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No

sudden hare
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ah no no

craggy sapphire
sudden hare
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yes yes

craggy sapphire
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No spoilers @subtle bough

craggy sapphire
sudden hare
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wait

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-x^2+10xy+23(1+23x)

craggy sapphire
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??

sudden hare
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nop

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wait

subtle bough
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-24x^2+10xy+23

craggy sapphire
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Don’t spoil it @subtle bough

subtle bough
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keep going

craggy sapphire
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They want to go though this on their own

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They’ve said this explicitly

celest fox
sudden hare
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-x^2 + 10xy + x((23/x)+23x

craggy sapphire
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@sudden hare you don’t have to overthink it

sudden hare
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mmmh

craggy sapphire
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You simply forgot to add the -23x^2

craggy sapphire
sudden hare
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just i think you knew too that i knew

craggy sapphire
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Hm it wasn’t clear

sudden hare
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yes

craggy sapphire
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This is where I wanted you to arrive

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Because now the problem is easier to solve

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If you complete the square

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You’ll see that you can reduce the problem even further

sudden hare
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and it is 22x^2

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ah no

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24

craggy sapphire
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Factor out -24

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Or wait

sudden hare
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oh 23 is 24-1 and i factorise

craggy sapphire
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You can make it nicer

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Yes

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Exactly

sudden hare
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oh nice i do that wait

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24((-x^2)+1) +10xy - 1

craggy sapphire
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Include 10xy aswell

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And then complete the square

sudden hare
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ok

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24((-x^2)+1+(10xy/24)

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-1

craggy sapphire
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Mhm

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Or -24(x^2 -5xy/12 - 1) - 1

sudden hare
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wait 1= x^2 + y^2 so i remplace that ?

craggy sapphire
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Sure you could do that actually

sudden hare
craggy sapphire
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So 24(y^2 + 5xy/12) - 1

sudden hare
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yes

craggy sapphire
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Now

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There’s different approaches here

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But let’s try the one that doesn’t require any special knowledge

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Just complete the square of y^2 + 5xy/12

sudden hare
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huh

craggy sapphire
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As in rewrite it in the form (a+b)^2

sudden hare
sudden hare
craggy sapphire
sudden hare
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not +

craggy sapphire
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According to whom?

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Can you say why?

sudden hare
craggy sapphire
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That was different because i factored -

sudden hare
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yes

craggy sapphire
craggy sapphire
sudden hare
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oh ok

craggy sapphire
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Since y^2 = -x^2 + 1

craggy sapphire
sudden hare
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no

craggy sapphire
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Have you completed the square before?

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Not interested

sudden hare
craggy sapphire
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Have you seen (a+b)^2 = a^2 + 2ab + b^2 before?

sudden hare
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yes

craggy sapphire
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I want you to try and go backwards

sudden hare
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but wait

craggy sapphire
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From y^2 + 5xy/12 to something of the form (a+b)^2

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Plus or minus a constant

craggy sapphire
sudden hare
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i add x^2 and -x^2

craggy sapphire
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Then we’re back again to your other simplification

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So not useful unfortunately

sudden hare
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mmh

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i add 25 and -25 ?

craggy sapphire
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Would u agree that (y+5x/24)^2 = y^2 + 5xy/12 + 5^2/24^2?

sudden hare
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it's 10xy/12

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oh ok

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i understand

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so yes i aggree

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so (y+5x/24)^2-5^2/24^2

craggy sapphire
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Mhm

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So

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Notice that

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This reduces the problem to maximising (y + 5x/12)^2

sudden hare
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why

craggy sapphire
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Set a = (y + 5x/12)^2

sudden hare
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yes ?

craggy sapphire
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Turns into 24(a - 5^2/24^2) - 1

sudden hare
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yes

craggy sapphire
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Is to maximise a

sudden hare
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OHHHH

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ok thanks

craggy sapphire
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Now you can do what u suggest from the start

sudden hare
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it's fnish i must derivate that and thats all

craggy sapphire
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By setting x = cos t, y = sin t

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Yes differentiate (sin t + 5cos(t)/12) ^2

sudden hare
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ah its easy i think

craggy sapphire
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Use chain rule for example

sudden hare
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yes

sudden hare
craggy sapphire
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Oops yes

sudden hare
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and i have to solve an equation when it's equal 0

craggy sapphire
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Mhm

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It might also be easier to approach the problem of differentiating differently

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Since you know ()^2 is behaved nicely

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So you might get away by studying the inside instead

sudden hare
craggy sapphire
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Im pretty sure you rewrite the inside into one sinus

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sin x + 5cosx/24 can be rewritten with trigonometry

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Before you differentiate

sudden hare
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yes exactly

craggy sapphire
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But also it’s sufficient to find when the inside is maximised

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So you can ignore ^2

craggy sapphire
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You’re done

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Which can even be achieved without calculus

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Just plain trigonometry

sudden hare
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idk how

craggy sapphire
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If you don’t know, then just differentiate and find max

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Of sin x + 5cosx /24

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Then square that max

sudden hare
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ahhh yes

craggy sapphire
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And put it in the expression from before

sudden hare
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Thank you for your patience and for taking the time to explain this to me.

celest fox
craggy sapphire
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Last year

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Or rather

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3rd year

celest fox
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Ok

mental arch
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Heyyy

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I am in 12th grade but ig ik how to solve it

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First make the above equation in just 1 term

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Either x or y

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Totally your choice

mental arch
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And then differentiate the equation

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And put it equal to zero

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And simplify to the value of x

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And use that value to get the value of Y

craggy sapphire
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Heyy

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Just use Lagrange multipliers

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Done

fossil rampart
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have you learned quadratic forms in linalg?