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.

[@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!!!!

[@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?

[@user-gf4vg2ey7g]

@@shireenkhan6847 what about this one? th-cam.com/video/LOwj7UxQwJ0/w-d-xo.html

[@user-gf4vg2ey7g]

@@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.

[@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.

[@awesomebearaudiobooks]

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

[@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.

[@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

[@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).

[@abdoennajah2702]

Finally i understood. Thank you for this great explanation

[@RayRay-yt5pe]

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

[@omniscienceisdead8837]

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

[@SaranshGupta786]

Very clear explanation, thanks a lot.

[@josephm276]

Excellent sir🌹🌹🌹🌹

[@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!

[@rishiraj5711]

Thanks for this.❤

[@bilenkeziban6237]

Well explained, thank you!

[@imaadnaeemansari8693]

Great explanation!!

[@nq2c]

this was super helpful thank you!

[@dikshabagh30]

really good video

[@rohan45]

Great explanation
Thank you so much

[@muzafferturgut6037]

Appreciated🙏

[@jayneabrown]

Super helpful. Thank you so much.

[@ouafaeraibi7847]

amazingly explained, Thank you!

[@sreyanjali]

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

[@jakobnilsson4704]

Awesome explanation

[@pratiksahu173]

I got this... 😊 Thanks a ton.. !!

[@swatisingh559]

This is really good⚘️⚘️

[@arwaalshikrian5354]

Extremely helpful

[@deeptysarder6797]

Nice graphical explanation

[@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.

[@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)?

[@hypebeastuchiha9229]

That was really helpful thanks

[@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?

[@williamgamelin154]

Well dons and thanks for memory boost.

[@devinahluwalia9289]

Was very helpful, cheers 🍺

[@haqseiitian]

thankyou bro...

[@sigmamathematics2215]

I got understand 1st time ❤❤❤

[@voyagers5026]

just great thanks a lot

[@hritikawadhawan6707]

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

[@viniciusbrilhante5112]

That was a really good explanation! Very clear!

[@shubrajchuckowree5670]

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

[@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

[@zainislam554]

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

[@primeview921]

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

[@mrak8948]

Grt explanation

[@albmrbo1]

Thank you!

[@dirkadrichem5468]

How about the sine and cosine function?

[@swatisingh3327]

❤❤❤❤wow

[@law9544]

What are u guys majoring in?

[@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

[@NNKNTA]

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

[@denvernarvasa3033]

thank you :)

[@JessaEstrada1315]

thanks!

[@solounomas0]

And why is this for?

[@iulianatoropoc5016]

Eureka! She cried.

[@handleyt2024]

legend

[@fragamus]

Exetra->et cetera

[@j.brightlee3314]

plz do more video plz plz

[@xhonimehmeti9432]

wasted 22 min of my life

[@indian_otaku2388]

wasted 10 seconds of my life