Find the local min/max of a cubic curve by using cubic "vertex" formula

Find the local min:max of a cubic curve by using cubic "vertex" formula,
sketch the graph of a cubic equation,
part1: • how to find the vertex...
part2: • vertex formula of a cu...
part3: • Find the local min/max...
blackpenredpen,
math for fun,
calculus for algebra students,
algebra-student-friendly calculus,

[@marmor2602]

I just love the fact that you can use the second derivative for choosing between minimum/maximum The concept is just so clever

[@blackpenredpen]

Marmorkuchen2 Thanks!!!

[@PerfectlyCrafted]

You are an amazing teacher. I love how I can show my friends who aren't in calc this stuff. Thanks!

[@blackpenredpen]

Perfectly Crafted it's my great pleasure!! Thank you!!!

[@orient9439]

BRO YOU ARE TRULY GREAT TO TEACH THIS CONCEPT IN A METHODICAL WAY

[@gerxi84]

Thank you so much! For some reason, all of my lessons tell me how to find the zeros but never the min/max! Thank you so much!

[@foreachepsilon]

You never have to plug in after this: plug in the “cubic” formula into the second derivative and you’ll get +/-sqrt(b^2-3ac) so the plus value will always be the local min and vice versa for the minus value.

[@Drytube]

But how? any proof or explanation would be appreciated.

[@Jonathan_Jamps]

Great explanation! Great video! I'll practice and try solve for X^4 by changing the polynomial degree.

[@tonypalmeri722]

I don't know whether someone else already mentioned this. There is no need for the second derivative to help you find which is the min and which is the max. Since the highest order term dominates, you only need to know the sign on (the coefficient of) that highest order term, and you can easily intuit the overall shape of the curve, and of course which of the extrema (left one or right one) is the min and which is the max.

[@joshcoe1992]

Thanks so much, you just really helped me out!

[@rgerk]

Suggestion for another video: How to calculate the roots of this equation.

[@TheBlueboyRuhan]

Great video as always!

[@dubble.]

I fully understood everything so you must've done a good job. Now I can use my newfound knowledge to impress the teachers at my exam, if I get this subject that is (this is outside the curriculum, so it should impress them).

[@cookiegod8510]

this is super helpful, thank you

[@luludolan7405]

how can i work out the values of a, b and c if my equation is factorised in brackets? do i have to expand the brackets before hand?

[@mukhammadrizomadjidov5136]

entertaining and relaxing video; i can't imagine how much effort is put into these videos ;)

[@mat5667]

Thanks for the vidjeo, was useful for solving an engineering problem!

[@NeillMikha]

Thank you! it is very helpful

[@reazraza]

I both love doraemon and math so this is the perfect channel

[@dylaninho2500]

Thanks so much !

[@JashanTaggar]

Love these man!, Back in Precalculus 12, I always wondered why the integral zero theorem worked. Could you please derive this theorem? (Prove it)

[@amrhammad2189]

Great ! Thanks a lot man 💙💙💙

[@gnikola2013]

Well, you technically don't need to do the second derivative test. Surely it is the easiest way to prove whether the value is a max or min (locally) but, if you think for a moment, you may realize that you can know the basic sketch of the function by looking at the cubic term coefficient (principal coefficient?). In this case, it is positive, so we know that the function increases, except in the interval (1;5) as they are the critical points we've just calculated, and as we know how cubic functions look like (something like an N), we can assure that the local maximum is at x = 1 and the local minimum is at x = 5.

[@MrRyanroberson1]

as nice as it is, this is for the rigorous form where it doesn't matter the degree as long as you have a vertex formula.

[@duckymomo7935]

Ryan Roberson it’s best to use second derivative test because first derivative test is useless in multiple variables or complex

[@zadramm1]

DO you think the X-axis is correct? i mean the graph does not pass the X-axis.

[@pinpinbowling4087]

Thank you very much for the video

[@thomasg6830]

There is an easier way to check, which of the x values is the local min/max. Just check the sign of 'a'. If its positive then the smaller 'x' is the local max, otherwise the local min. The larger 'x' will always be the opposite. All because: cubic formula.

[@charleskennethvirrey1439]

damn, you really does look like friendly. Your tone and facial expression, helps a lot from understanding the problem. You doesn’t look like one of those terror perofessors

[@morrowseer7737]

How this man teach more than my math teacher in 9 minutes

[@jchry3688]

How did you even read my mind Really I was thinking about local min/max for cubic polynomial 10 minutes before you uploaded this I'm lost in word...

[@axemenace6637]

Atomizer The Law of Large Numbers, Sir.

[@sunilparekh4581]

YT algorithm is about to read your mind be safe 🤣🤣🤣🤣

[@tyronekim3506]

Good video. I would like to know the method used to find the three roots at y=o. Thanks.

[@johnrodonis4186]

No need for second derivative...especially in a precalc class where it hasn't been taught yet. However, in a cubic where "A" is +, the SMALLER "critical" value (in this case, 1) MUST give the MAX and the larger (in this case, 5, the MIN, because of the behavior of the cubic function. IF "A" is negative, the opposite is treu.

[@artsysahelcherry2269]

Thank u so much

[@bbadhum6189]

Thank you :)

[@ivanlara8024]

Good video!!

[@bezelnarrow3799]

Man I know u r god tier as soon as I heard that doraemon theme song. Ty uwu !

[@georgiaoshea5678]

Literally saved my life

[@overlordprincekhan]

Hey bro, I am a student, I would like to ask you a question. What will I do if the square root of the quadratic formula in the video becomes imaginary term? Please, I would be pleased if you answer to my question

[@coldguy]

Finally I can find and graph a cubic function

[@reetsempire5504]

Thanks

[@jadepangwaiho]

Hello, may I ask you an interesting question? The domain of x^0 is R\{0} isn't it? However, wolfram alpha and symbolab show me that the result is R, can you explain which of the answer is true? Thank you so much .

[@ninadee9753]

Is the second derivative able to be used for any cubic equation?

[@TheMiningProbe]

I love your videos so mucb

[@blackpenredpen]

best blader thank you!!!!!!!

[@reazraza]

Intergration of (awesome video) limit 0 to 1 ☺

[@Chai_yeah]

Great , but don't you need roots of eqn to sketch the graph more accurately coz here The min at x=5 can also be zero or a positive no. ( here it is indeed negative but still)

[@anacarolinaorlandin5569]

Hey! I have a question: what happens if you dont have one of the 'a, b or c' values? The cubic function I'm analysis doesnt have the c value, only the 'd'. Should I just use 0 instead? How does it work?

[@triple7marc]

Yes, it would be 0, because any time it's 0 it would just cancel out the entire term instead of you having to write "0x."

[@ujjwalkarnani1059]

Bro that intro music is nostalgic

[@keyurd.6315]

you should have probably solved the equation for the roots, to know where the curve will cut x axis so as to get more accurate graph

[@kaeri4537]

Does it work with 2x^3-5x^2+4?

[@denisnyantori3661]

Thanx

[@rith359]

Now that I just learned this, give me a quiz

[@BP-td5gv]

When I put this question into the calculator is says X1 = 6.656 X2 = 2.577 and X3= -23319 The setting was Polynominal 3

[@ijabbott63]

Rather than testing the 2nd derivative at the vertices, why not just check the sign of 'a'? If 'a' is positive, the vertex with the lowest 'x' coordinate will be a local maximum and the vertex with the highest 'x' coordinate will be a local minimum. If 'a' is negative, they are the other way round.

[@jaykaushik1775]

I agree with

[@adamkangoroo8475]

Deja vu, I've just watched this math vid before

[@user-vw8wl3pl7h]

how if the b is a possitive number and i get a negetive square root

[@mohammadumair8636]

Check will be given or not..how check comes from

[@MrQwefty]

Hey BPRP, try to solve this WITHOUT L'Hôpital: The limit as x approaches pi/3 of (1-2cos(x))/(x-pi/3)

[@blackpenredpen]

MrQwefty hmmm I assume I can't use the definition of derivative neither right?

[@MrQwefty]

Why not? :)

[@blackpenredpen]

Lol. Let t = x-pi/3 and use well known limits (cosx-1)/x and sin(x)/x go to 0 and 1, respectively, as x goes to 0.

[@kgshbteambeasto_o963]

DORAEMON! also BPRP, can you solve this problem? define f(x) as following 1 if x is rational 0 else what is the area covered under f(x) in [0, 1]?

[@blackpenredpen]

Oh, Dr. P did it already: kzbin.info/www/bejne/oHPPZKGpnJWlnsk

[@kgshbteambeasto_o963]

thank you! And can you make a video on cardinals :D i.e. aleph_{n}

[@attention8312]

I don't know if anybody will ever see this but why would you do this instead of differentiating the cubic function so you can sub it into the minus b formula? It seems more complicated than it has to be doing it this way

[@mehulmanian]

So it’s just the zeros of the first derivative? Kinda seems like more work tbh

[@pawananubhav12]

love the doraemon theme 😃

[@bloodstaindeyes0420]

holy fuck you saved me i have a test in 3 hours and i have not slept in 23 hours

[@jaykaushik1775]

Snejpu very good

[@sinom]

Interesting and all but I don't think it's much faster than getting the first derivative and plugging that into the quadratic formula.

[@helloitsme7553]

Sinom well that's exactly what he's doing though.

[@ateium2409]

Is it just me or there is a music kind of like the Doraemon Theme song at the Beginning and Throughput the video???

[@tomctutor]

Why not just solve dy/dx = 0 the traditional way (unless I am missing something here), as interesting as your video is, I think it may confuse some students!

[@snejpu2508]

Interesting, but why not derivatives?

[@AndDiracisHisProphet]

well, he derived the vertex formula using derivatives, so...

[@jaykaushik1775]

Yes you are correct !

[@AndDiracisHisProphet]

what a surprise

[@jaykaushik1775]

Thanks

[@Golden_Tortoise]

Why is the second derivative not 6x-18 ??? How did he get 6x + 2 ???

[@sontubanerjee9949]

You can teach IITJEE Mathematics.

[@khaled014z]

but why is he considering the zeroes of the function itself rather than the zeroes of the first derivative?

[@admink8662]

you must watch the videos in description

[@gnikola2013]

He isn't, that formula is not for finding the functions zeroes, watch the videos in the description

[@admink8662]

It's not the formula for finding the function's zeroes. Actually it's the formula for finding the the first derivative's zeroes.

[@tarat.techhh]

got here first YAY!!!!!!!!!!!!!!!!!!! blackpenredpen

[@Apollorion]

Can anyone here proof that the minima and maxima from real polynomial functions of an odd degree are always local?

[@Metalhammer1993]

no proof, cause i have no idea how to proof anything but my own mathematical incompetence, but it is logical right? cause in general they do not have maxima. outside of a reasonable scale the x^3 part will just become dominant. like take a polynomial of any degree really. plug in x=1000000000 or something over crazy. you´ll see the only relevant part is the highest degree. so you just take the limits. outside of the area you´re looking at these things just race to their limits. However even degree polynomials have the same limit for positive and negative infinity. meaning it leaves a real minimum/maximum (depending on the sign of the highest degree). Odd degree polynomials however DO cover both extremes. they havbe a negative limit anda positive one while the even degrees only cover one. Meaning if you just go far enough either to the left or to the right you´ll get below or beyond your local maxima you´ll send your polynomial right into either negative or positive infinity. now somebody with better mathematical education than me: make a rigoros proof out of that or tell me the name of one that says mathematically what i described verbally^^

[@turbopotato4575]

Its basically what Metalhammer1993 said. To put it more rigorously consider the polynomial p(x)=a(n)*x^n +a(n-1)*x^n-1 + ... + a(0) where a(n) are constants and n is odd. It can be factored as x^n *(a(n) +a(n-1)/x +.... + a(0)/x^n) for big enough x a(n)x^n is a good enough approximation since all a(k)/x^k tend to zero case 1 a(n) > 0 then lim x-> inf a(n)x^n = inf so it has no global maximum and lim x-> -inf a(n)x^n = -inf so it has no global minimum. the argument is the same for a(n)

[@foreachepsilon]

Prove that lim x->+/- infty (cubic) = +/- infty. This means that there exists an M > 0 s.t. For all x, abs(f(x)) > M. Meaning M + abs(f(local max/min)) > abs(f(local max/min) proving the local max/min is never an absolute max/min.

[@Apollorion]

Eh.. turbo p, I think where you -said- eh.. _typed_ "local" you actually meant something like "absolute" or "global".

[@Metalhammer1993]

thanks turbo potato

[@johnclemash2544]

Great AI is now ruining shorts now 😢

[@shubhaminfinity1966]

❤ from indians

[@OonHan]

0 views! yay