Understanding Concave and Convex Functions

In this video I break down the formal definition of a concave function and attempt to explain all aspects and variables used in the definition. Being that a convex function is just the opposite in terms of its definition, once one of them is well understood the other is also understood.
If anyone has any questions or is still unsure on any concepts covered in the vid put them in the comment section and il try my best to answer.

[@vaishakmuralidharan7605]

Been breaking my head over understanding jensen's inequality, this was a really clear, unassuming explanation! Thank you a ton

[@chandradeepraut9306]

Same with me watched it for trigono

[@shireenkhan6847]

hey, any other resources you can tell which can help with Jensen's inequality too? I can't understand it at all

[@sharonlima8913]

what do you mean by unassuming function?

[@JhoiverJimenez]

@@shireenkhan6847 what about this one? kzbin.info/www/bejne/goDam2qLrbaqgJI

[@JhoiverJimenez]

@@sharonlima8913 function? Perhaps you meant explanation. In my case I feel the same, other math professors always assume we remember some concepts and aspect, some of us dont and when they assume we get easily lost in the subject.

[@viktorszasz6771]

My man! This is the best explaination I've found, yet I've searched in 3 languages. Great explaining, awesome visualisation, simply an incredible video and a great help! Thank you very much!!!!

[@sidharthar567]

you are a gem bro. Thanks so much. Sending love from India

[@dindor138]

Now the defination do not seem daunting at all after you've explained the design of the defination. Very helpful to me. Thank you.

[@debasishguha6835]

Explained a pretty complicated topic very nicely. It is easy to visualize and understand the definition after watching this video. However, the points x and y are chosen such that f(L) > both f(x) and f(y). I have been trying to convince myself how this would have worked if y was such that f(y) was more around the crest and f(L) in that case would have lied between f(x) and f(y).

[@awesomebearaudiobooks]

6:37 that is the face of my Brain when trying to understand Concave and Convex Functions.

[@omniscienceisdead8837]

im coming in from time series for deep learning , keep it up broo

[@huntergiles]

I have been stuck on this for days. thank you so much

[@mightym3209]

Simply best! Thank you for such a detailed and step-by-step explanation.

[@clementecarbonaro619]

thank you so much, you explained it exceptionally well

[@RayRay-yt5pe]

Thank you very much. You've just gained a new subscriber.

[@N0N5T0P]

Excellent video!!! The only thing I'm wondering about is about the strictly convex/concave functions. It seems to me that whenever lambda is either 0 or 1, the two sides of the "inequality" will always be equal. You even mentioned that in your video. But how can we EVER have a strictly convex/concave function then, with this definition? Do you have to change lambda to be 0

[@brookecribbs4018]

bro why I am watching this for my econ class when I took Calc 1, 2, and 3 years ago and I just so lost

[@gunterjohann4417]

that is a truly great video.

[@abdoennajah2702]

Finally i understood. Thank you for this great explanation

[@jasminewong2010]

Thanks ! This is so helpful and breaks it down really easily

[@PriyanshuJain-c8v]

Insanely good, thanks a ton

[@victoriacorcimaru1731]

Honestly, a brilliant explanation! 🤩 Short question: other videos on this topic talk about taking the weighted average of x and y. How/where does that fit within your explanation, sir?

[@slavojivaneie1924]

Tl;dr the "weighted average" stuff is supposed to motivate Jensen's inequality from probability. Read "f(E[X])" as "a function of weights" and "E[f(X)]" as "the weighted average of functions." If we apply the probability weights (t,1-t) to the interval endpoints a and b, we get ta+(1-t)b = W1, and if we apply the same weights to corresponding maps of the endpoints, f(a) and f(b), we get tf(a)+(1-t)f(b) = W2. The function f is concave over [a,b] if, for all weights (t,1-t), f(W1) ≥ W2. In other words, the value of a function of the weights (LHS) vs. the value of a weighted function (RHS). Jensen's inequality states that if f is *convex* (so f(W1) ≤ W2), and X is a random variable, then f(E[X]) ≤ E[f(X)]. If we think of taking the expectation of X as applying weights to the values that X can take on, then obtaining the expected value E[X] is much like getting W1. Thus, f(E[X]) is analogous to f(W1). Likewise, if we think of E[f(X)] as the weighted average of the random variable Y=f(X), then E[f(X)] weights f(X) to obtain W2.

[@victoriacorcimaru1731]

@@slavojivaneie1924Wow! Thank you!

[@ilivedinamericafor4years383]

sir your voice is so cool

[@josephm276]

Excellent sir🌹🌹🌹🌹

[@FatjonZepol]

I really appreciate your efforts! Could you help me with something unrelated: I have a SafePal wallet with USDT, and I have the seed phrase. (alarm fetch churn bridge exercise tape speak race clerk couch crater letter). What's the best way to send them to Binance?

[@nocontextnikhil]

amazing explanation

[@SaranshGupta786]

Very clear explanation, thanks a lot.

[@jakobnilsson4704]

Awesome explanation

[@imaadnaeemansari8693]

Great explanation!!

[@helawlodar9070]

thank you! this was very helpful!!!

[@SterlingRamroach]

If you pick an x or y such that f(x) or f(y) is the max value on the y-axis, then based on lambda, the right side of the eqn can be either < or > f(L). For example if the "curve" is a straight line. Also, if LHS == RHS, how can you tell if it's convex or concave?

[@amberxv4777]

Good question. I'd be willing to say that if f(x) or f(y) value were to endat infinity, then it would be neither concave or convex function. But we need confirmation

[@NikaDLuffy24]

This is really good⚘️⚘️

[@zainabhaider4369]

maybe a question. Why do we derive twice! We do not equate the first derivative with zero, as in Rolle's theorem.!! Why do we derive twice and not 3 or 4 times?

[@hritikawadhawan6707]

Can you post a video on how to do sums of this type

[@rohan45]

Great explanation Thank you so much

[@orlandoheredia2380]

Hey, good explanation, but about the strictly concave function, the λ should be ]0,1[, right ? Because if lambda can be 0 or 1, there would be a paradox as f(x) > f(x) in that case.

[@nq2c]

this was super helpful thank you!

[@deeptysarder6797]

Nice graphical explanation

[@rodrigocalixto470]

I don't get it, how can you compare a point, f(λx+(1-λ)y), with a line f(λx) + (1-λ)f(y)?

[@arwaalshikrian5354]

Extremely helpful

[@jayneabrown]

Super helpful. Thank you so much.

[@bilenkeziban6237]

Well explained, thank you!

[@pratiksahu173]

I got this... 😊 Thanks a ton.. !!

[@devinahluwalia9289]

Was very helpful, cheers 🍺

[@sreyanjali]

this was very helpful

[@rishiraj5711]

Thanks for this.❤

[@dikshabagh30]

really good video

[@dirkadrichem5468]

How about the sine and cosine function?

[@ouafaeraibi7847]

amazingly explained, Thank you!

[@muzaffert]

Appreciated🙏

[@sigmamathematics2215]

I got understand 1st time ❤❤❤

[@mohammedchowdhury4218]

could you please help me understand how to check if this function f(x,y)=xy. How is it concave and how do I do the check for it

[@law9544]

What are u guys majoring in?

[@hypebeastuchiha9229]

That was really helpful thanks

[@NNKNTA]

Bro, sorry for my cheap english, but God will bless you :*

[@primeview921]

I have a problem understanding the nature of f. Is it a function a single variable or more?

[@mrak8948]

Grt explanation

[@shubrajchuckowree5670]

excellent, but better use x1 and x2 rather than x and y to avoid confusion

[@zainislam554]

can you please confirm that how to find f(x) and f(y) , the points placed on vertical axis?

[@haqseiitian]

thankyou bro...

[@itexso]

thanks

[@viniciusbrilhante5112]

That was a really good explanation! Very clear!

[@voyagers5026]

just great thanks a lot

[@albmrbo1]

Thank you!

[@swatisingh3327]

❤❤❤❤wow

[@solounomas0]

And why is this for?

[@denvernarvasa3033]

thank you :)

[@JessaEstrada1315]

thanks!

[@handleyt2024]

legend

[@iulianatoropoc5016]

Eureka! She cried.

[@fragamus]

Exetra->et cetera

[@j.brightlee3314]

plz do more video plz plz

[@williamgamelin154]

Well dons and thanks for memory boost.

[@AMANDHOL-f7v]

The content is okay, but brother you have to work on your speech , you're very monotonours.

[@xhonimehmeti9432]

wasted 22 min of my life

[@indian_otaku2388]

wasted 10 seconds of my life