2 questions in 2 pictures

[devout copper]

All description is in the photos

Anyone who solves these doubts for me , will be called sir by me for the rest of my life

I am only pinging cause akki gave me permission , @polar flare here

Please say yes if you agree

So that it remains as a proof.

[polar flare]

I agree to this

[devout copper]

Now , the above 2 pictures contain 2 doubts seperately

1 is algebra and another one is my own kinda theorem thingy calc + inequality

[polar flare]

The proof one seems right, I suck at proof part tho

[devout copper]

It seems correct ?

Can i solve inequalities that way ?

And regarding the algebra one ?

[polar flare]

Yup

[polar flare]

Unless

The function is a piecewise function

[devout copper]

Yeah , then its a pain

[polar flare]

Yee

[devout copper]

Algebra one now

[polar flare]

How did you reach 2^n + 4 ?

Doesn't integral solution mean all x values where y is an integer ?

[devout copper]

Its was like a torn out process of thinking cause what regularly happened is that denominator would have a factor which would not cancel out as numerator only contains 2s mainly , thus i thought that maybe i can make the denominator be 2^n

[devout copper]

All integral values of x for which y is an integer

[devout copper]

That worked splendidly as i found 4 solutions

But then i found out there are 2 more which i couldnt find

[polar flare]

Denominator 2^n
numerator 2^{n+3} + 2^5

[devout copper]

What

[polar flare]

Wait imma try out some values brb

[polar flare]

I multiplied 8 inside

[devout copper]

One thing to notice is that x is 2^n+4

So what happens is

8*(2^(n)+4)/(2^(n))

You can apply n is the above equation

[polar flare]

Yee I m simplifying that that

[polar flare]

In this

[devout copper]

Okay

n =2,3,4,5 works

I tried 6,7 but it didnt work

So i stopped

[polar flare]

Try 8+2^{5-n}

Oh that's why it stops at 5

Hmm

[devout copper]

Idk where the other 2 are

[polar flare]

Is there any other way we can keep it integer ?

Wait there is

We just need to keep

2^{n}+4 an integer

[polar flare]

While keeping this one too

So for an n where it cancels numerator ?

[devout copper]

Yup

[polar flare]

After this our numerator will be odd

But denominator will always be even ?

Denominator

[devout copper]

Yup

[polar flare]

Hmmmm

[devout copper]

No

denominator wont be always even

[polar flare]

Ohh when will it be odd ?

[devout copper]

When x is odd

[polar flare]

Denominator in this

Ot in original one

When we take n >5

[devout copper]

original one

[polar flare]

It will be 2^-{n-5}

[devout copper]

denominator wont be odd if you are using my approach

[polar flare]

Ahh that won't work we need some other way

[polar flare]

This is that approach 😂

[devout copper]

I made my approach specifically so that denominator always stays even

[polar flare]

I m still using your approach

[devout copper]

Then denominator cant be odd

[polar flare]

Yee

We need new approach i guess ?

[devout copper]

yup

new approach to find the remaining 2

[polar flare]

Numerator is 8x so we will need it to be even

[devout copper]

finding an approach is so hard in this question

[polar flare]

Ikr

What if

[devout copper]

See , thats why i kept denominator even

[polar flare]

We work backwards ?

[devout copper]

You mean x = y something ?

[polar flare]

We find answer

Then find approach ?

[devout copper]

You gotta bruteforce then

[polar flare]

Yup

[devout copper]

Then there are 2 independent values

[polar flare]

And then we try to make them not dependant , like find a logic in between

[devout copper]

And 10000 possibilities if we limit x and y between 0 and 100

I would try my best to not bruteforce

[polar flare]

Hmm brute force ain't that good of an idea after all

[devout copper]

Yeag

I have jee adv in 2 years

[polar flare]

Hmm

8

[devout copper]

I am kinda shaking in my boots

[polar flare]

8*8/4

It is an integer

Then 12

8*12/8

Still integer

[devout copper]

I already found those

[polar flare]

Cuz they cancel each other

[polar flare]

2,3,5,8,12

We have 5, we need one more

[devout copper]

Wait wtf

[polar flare]

20*8/16

[devout copper]

how 2 ?

[polar flare]

Oh ye it's negative

[devout copper]

Are you giving the values of x ?

[polar flare]

I was

But it's negative

My bad

[devout copper]

Not 3 aswell

Next one is 20

x = 20

[somber badger]

Hi please DM me if you need help

[devout copper]

I found that

[devout copper]

Please try to help here if you can

as akki and me , both can learn from your techniques cause its kinda hard for us

we both are preparing for exams

[polar flare]

Ok x values we have 5,6,8,12,20

[somber badger]

I like dealing with individuals based on my expertise. At times I may need just a little tokens. You can DM if you also need help with your exams. Thank you!

[devout copper]

5 doesnt work i think

[polar flare]

X value ?

[fathom saffron]

You can't charge in a group chat

[polar flare]

5 works

[devout copper]

X = 5

[polar flare]

Cuz denominator 1

[polar flare]

Why not

Oh wait yes

I forgot

[devout copper]

What , why not

[polar flare]

You can't charge in this server

[polar flare]

All free help

[devout copper]

Whats charge

[polar flare]

Good job Kanny

[polar flare]

Ask for monee

Krypton

36

[fathom saffron]

[devout copper]

Ohhh , money for advice

[polar flare]

9*8*4/8*4

[devout copper]

sorry , no , i prolly have some better places to spend it

[polar flare]

We have 6 values, gotta make a connection now

[somber badger]

ok

[polar flare]

5,6,8,12,20,36

How to make the connection now

2^n + 4 = 36
n = 5 here

[devout copper]

Wait

[polar flare]

We did till n=5

[devout copper]

How 6 ?

[devout copper]

Yeah

[polar flare]

8*6/2

X value

[devout copper]

Okay

[polar flare]

It doesn't follow 2^n +4

[devout copper]

x = 24 ?

[polar flare]

It does

All do

[polar flare]

8*8*3/4*5

Nope

5 and 3

Brb

[devout copper]

Yeah , i was thinking that how

[polar flare]

We have 6 values tho

Now we need connection

5 = 2⁰+4

6=2¹+4

8=2²+4

12=2³+4

20=2⁴+4

36=2⁵+4

@devout copper you are a genius

[devout copper]

5 doesnt work

[polar flare]

Your solution was perfect

[polar flare]

Why not ?

8*5/1

[devout copper]

i dont think even 6 does

[polar flare]

8*6/2

It works see

[devout copper]

Wait , lemme think

[polar flare]

They are integers

[polar flare]

Why think we have the answer

Your solution was perfect

It was just half done

2^n+4

[devout copper]

Wait

[polar flare]

How many constraints did we have ?

It has to be greater than -1

Or better

Greater than or equal to 0

Smaller than or equal to 5

Cuz 8+2^{5-n)

We can't have less than 0 cuz 2^n+4

We just had to make 2 inequalities

[polar flare]

@devout copper want me to explain ?

We just need to make 2 inequalities

[devout copper]

Wait

Holy shit

I cant believe this

[polar flare]

You are brilliant

[devout copper]

What the fuuuuuuck

[polar flare]

You just didn't see things completely

[devout copper]

I am not

[polar flare]

Krypton you are the smartest 👍

[devout copper]

Nah man

[polar flare]

You solved the question ages ago

[devout copper]

It is still a coincidence

[polar flare]

Is it ?

Want me to show you why it is not ?

[devout copper]

No but my approach at 2^n+4 was like that

Like , coincidence

[polar flare]

you did find your approach yourself right ?

[devout copper]

Yeah

[polar flare]

after finding the approach we just make 2 constraints

[devout copper]

Yeah

[polar flare]

$x=2^n + 4$ we find that n must be greater than equal to 0

[dense patrol]

Akki

[polar flare]

with $y = 8 + 2^{5-n}$ we find n must be less than equal to 5

[dense patrol]

Akki

[polar flare]

voila your approach was not a coincidence

[devout copper]

I didnt understand this tho

[polar flare]

ohh

$y = \frac{8(2^n + 4)}{2^n}$

right ?

[dense patrol]

Akki

[devout copper]

Yes

[polar flare]

$y = \frac{2^32^n + 2^32^3)}{2^n}$

[dense patrol]

Akki

[polar flare]

just took 8 inside

[devout copper]

Yea

[polar flare]

$y = \frac{2^{n+3} + 2^{5}}{2^n}$

[dense patrol]

Akki

[polar flare]

now 2^n goes up

[devout copper]

Yes

[polar flare]

$y = 2^{n+3-n} + 2^{5-n}$

[dense patrol]

Akki

[devout copper]

Okay

[polar flare]

$y = 8 + 2^{5-n}$

[dense patrol]

Akki

[polar flare]

now we have this

[devout copper]

Ohhhhhhh

Fuckkkkkk

Ooooooooooh myyyyyyyy gaaaaawd

[polar flare]

I was using your approach all along

[devout copper]

Fuck man

[polar flare]

thats why I said you are a genius

[devout copper]

I didnt think my approach had this much potential

[polar flare]

from this we know n must be less than or equal to 5

[devout copper]

Yeah

[polar flare]

now wanna know a little secret 😂

[devout copper]

Yeah ?

[polar flare]

I did brute force for all the solutions ☠️

before realizing

[devout copper]

I see

Holy shit

[polar flare]

how did you come up to x = 2^n + 4 tho ?

[devout copper]

I still cant believe my ass

[polar flare]

smart ass

[devout copper]

I wanted 2^n form in my denominator so that 8x in numerator can easily cancel my denominators

i am fucking exhausted

[polar flare]

ohh okie go rest

imma interogate your solution till then

[devout copper]

5-6 hours self study and 9-3 coaching classes

[polar flare]

and me here with full day time pass

[devout copper]

All in all 11-12 hours

[polar flare]

I will start studying tomorrow 😤

[devout copper]

And at the end of the day , this gets me fucked up xD

[polar flare]

issoke issoke

[devout copper]

glad that i figured it out

[polar flare]

it will pay off well

[devout copper]

Yup

Definetly

[fathom saffron]

Both preparing for jee?

[devout copper]

Yup

[polar flare]

yup

mine is next year tho

[devout copper]

I have jee in 2 years

[polar flare]

ohh

[fathom saffron]

Nice

[polar flare]

whaat

[fathom saffron]

Goodluck

[polar flare]

I thought your adv was this year 😂

[devout copper]

Fuck , i wanna get into an iit so hard

[devout copper]

Nah

[fathom saffron]

[polar flare]

11 hours in 10th ☠️

[devout copper]

I just passed 10th

[polar flare]

oh so 11th

yee I passed 11th I forgot

[devout copper]

Gave board exams like 1 month ago

[polar flare]

so you have learnt calc ?

[devout copper]

I am gonna raise to 12 hours

And 13-14 in 12ths

[polar flare]

I m barely keeping up with 1 hour

[devout copper]

Yeah

Kind of

[polar flare]

I mean I have my reasons 😤

[polar flare]

oo good

[devout copper]

I dont know integration

[polar flare]

integral too ?

[devout copper]

I know limits , differentiation

[polar flare]

are you working in same pace in phy and chem too ?

[devout copper]

Yup

[fathom saffron]

Also in the question 1, is f'(x) < g'(x) is part of question?

[polar flare]

another question

I think its part of solution

[fathom saffron]

Hmmmm

[devout copper]

It was part of another question in which i tried to present a general solution to general inequalities using calculus

[fathom saffron]

I see

[devout copper]

please do comment on it as you like it

I am open to feedbacks cause i am still learning

[fathom saffron]

I'm just making sure you didn't mean if f(x) < g(x), implies f'(x) < g'(x)

[devout copper]

Nope

[fathom saffron]

Alright

[devout copper]

Both are required independently

[fathom saffron]

Ok get it

[devout copper]

I think i should close it now

@polar flare dm if you want , i am gonna sleep

[polar flare]

Oo okie

[devout copper]

+close