help please give me a hint atleast

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[high dirge]

Do you know about convexity?

[gusty forge]

We can directly assume a simple function and solve

[gleaming galleon]

yes

[oak summit]

The question has us compare f(y) with x, and f(y) + f'(y) seems very relevant to f(y+1)

[gleaming galleon]

i assumed g(x) = g(x+1)-g'(x) - x

[gleaming galleon]

yee

[gusty forge]

Mein kya bolna chahta hu ki. Direct fx is -6x^2 leke prove kar de

[gleaming galleon]

then how do we handle the y

[high dirge]

f seems to be concave here

So what can you say about the first order definition of concavity

[gleaming galleon]

it will be a decreasing function?

i mean f'(t) is a decreasing function its given right

[high dirge]

Yes but not my point

[gleaming galleon]

any concave function defined on an open interval is continuous?

[high dirge]

An inequality

[gleaming galleon]

suppose x<y<t

then f(t)-f(x) / t-x >= f(t)- f(y) / t-y

also how are u recongnizing it as a concave function can u please tell

[high dirge]

f'' < 0

Tjat is the second order def of concavity

[gleaming galleon]

oohh okayy

then what do we have to do

[high dirge]

The first order definition

The linear approximation is higher than the functin

[gleaming galleon]

wait shouldnt it be lower?

lower than the function?

[high dirge]

that's for convex functions

[gleaming galleon]

convex higher

yea thats what im saying

or am i confused w the shapes

[high dirge]

for a fixed y, f(y+t) < f(y) + f'(y)t

the RHS is the linear appprox

[high dirge]

no we're not talking about inner chords

we're talking about slopes

so the tangent line is always above

[gleaming galleon]

hi can u once hop on a call and explain it to me

please

[high dirge]

no

[gleaming galleon]

oh okay :/

[high dirge]

Sorry

Anyway, what I'm saying is the following

[gleaming galleon]

its okay

[high dirge]

if you take the curve of a convex function

a tangent at any point will always be below the curve

[gleaming galleon]

yes yes

but for concave the tangent will be higher

[high dirge]

the opposite for a concave function

no

[gleaming galleon]

sorry mb

[high dirge]

for example, take the function f(x) = x²

[gleaming galleon]

yea yea i got it rn

[high dirge]

you can draw a tangent at any point, it will always be below the parabola

[gleaming galleon]

yes yes i got it now

[high dirge]

on the other hand, for f(x) = -x², it's concave, so the other way around

So the first order definition for convexity is as follows

[gleaming galleon]

then we show that the linear appromixation will be higher than the function?

and how can we rigorously show that, like i got the idea but idk how to show it

[high dirge]

$\forall y, \forall t, f(y+t) < f(y) + f'(y)t$

[viscid rapids]

Rion

[high dirge]

Well this is just an equivalent definition to strict concavity

you don't have to prove it, but if you want to, there's always a way

[gleaming galleon]

oh and can u once explain what this represents

like what points the y and t are

[high dirge]

y is the point of tangency

t is the displacement

[gleaming galleon]

ohh okayy

and then from this how do i prove my original question

[high dirge]

Well here you want a very specific displacement

[gleaming galleon]

like?

[high dirge]

1

[gleaming galleon]

so t = 1?

[high dirge]

apply this to the y given in the exercise

and t = 1

https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=83

Page 69 explains the first order definition and provides a proof if you're interested (equation 3.2)

[gleaming galleon]

okayyy

wait ill once do the

sum and show it?

[high dirge]

Well for now just admit that it is true

we can come back to it later

[gleaming galleon]

okayy

[high dirge]

You can leverage this to show what you need to show very fast

[gleaming galleon]

im applying t =1 on that

[high dirge]

yes

so what do you get

for the specific y given in the exercise

[gleaming galleon]

f(y+1) < f(y) + f'(y)

[high dirge]

there's a leftover t there

change it to 1

yes

good

on the other hand, what else do you know about f(y+1)

[gleaming galleon]

greater than f'(y) + x

[high dirge]

Yes

Now what do you infer?

The conclusion should be immediate here

[gleaming galleon]

we get that

f(y) + f'(y) -f''(y) > x

after this?

[high dirge]

where did the f'' come from

[gleaming galleon]

wait nvm

give me one min

oh

yes

it was a

silly mistake

i accidentally wrote doube derivative

in my exercise book

[high dirge]

so

[gleaming galleon]

yea so it got proved

also is there any other place

for the first oder definition

like only the concave one

not the convex

its kinda confusing

real analysis is new to me

[high dirge]

Well a function is concave when its opposite is convex

[gleaming galleon]

yea yea but like a solid proof of concanve would be more convininent

[high dirge]

you just apply everything to -f

since -f is convex

[gleaming galleon]

also what does the inverted delta sign and the T mean?

[high dirge]

It's the gradient transposed

in one dimension it's just f'(x)

[gleaming galleon]

oh okayy

also how can we assume the t to be one

like why cant it be a variable

is it because its that fixed point on the tangent?

[high dirge]

What do you mean

[gleaming galleon]

like how can we assume the point t to be fixed

[high dirge]

Well we know it is true for any y and t

so you just pick the y and the t that suits your needs best

[gleaming galleon]

oh okayy

also

how can i approach this one

oh haha nvm i think i can do this

i didnt see the 24

[gusty forge]

@high dirge it's the nabla opertor right? Also used in schrodinger wave eq

[high dirge]

wdym operator, it's just differentiation

[gusty forge]

Yeah. It's just differentiation. In all the 3 coordinates

It's called that I believe

[gleaming galleon]

@high dirge i had a question

for the number of polynomials

why do we have to do find the coefficient

of that particular x^n

like whats the intuition behind finding the coefficient of a GP

[high dirge]

of what now?

[gleaming galleon]

for example

[gleaming galleon]

i was doing this sum

and the number of polynomials = coefficient of x^2020

in the GP

[high dirge]

what sum?

[gleaming galleon]

this one

[high dirge]

To be fair I don't even know how you got there

or what you're doing

[gleaming galleon]

https://math.stackexchange.com/questions/1964193/finding-coefficient-of-xk-in-a-product-of-two-polynomials

something like this

[gleaming galleon]

@high dirge another approach

id say its easier

[high dirge]

I guess that works