INEQUALITIES Full Chapter 🔥 | Triangle AM ≥ GM | Lecture 2 | JEE Foundation & Olympiads

追加: 2026-06-01

[00:00:12]Hi everyone.

[00:00:21]>> Hi everyone.

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[00:01:11]H S H S H S H S H S H S H S H S H S H S HP with the harmonic mean harmonic progression progression harmonic mean harmonic progression AMG arithmetic mean and geometric mean hello hi hi neuron 22 hello how are Is absolutely fine. Session session.

[00:01:45]Good evening neuron 22.

[00:01:53]session.

[00:01:57]Hi, back after long.

[00:02:03]We're back after long.

[00:02:35]Shall we begin the session?

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[00:02:43]Hi Roy. Hello.

[00:02:49]attendance.

[00:02:51]Live watching like all neuron. No issues.

[00:03:03]No issues.

[00:03:08]What's your name? Neuron YouTube.

[00:03:14]What's real name?

[00:03:26]Oh, hi V. Hello.

[00:03:39]>> Okay, let's begin the session better.

[00:03:42]Hi everyone. So, I welcome you all in the VSant science channel. I am your educator AL. You can call me A B A. And today the very interesting topic for the Olympia level for the foundation level that we are going to study the AMGM.

[00:03:57]This is lecture inequality. Inequalities how we use inequalities formated question. What is the modular function?

[00:04:05]Mod function related problems.

[00:04:09]Right? In this lecture we would be learning the fact am or gm right starting the session a very good motivation the beautiful thing about the learning is that no one take it away from you learning that is that we call asan knowledge so keep on sharing the knowledge keep on giving knowledge.

[00:04:48]That is the greatest virtue that no one take it away from you. Learning this is the strategy I'm following.

[00:05:06]Right. Right.

[00:05:10]This is the channel founding the topic.

[00:05:33]What is the meaning of a.m.

[00:05:39]9 a.m.

[00:05:41]10 a.m.

[00:05:45]6 a.m.

[00:05:51]right?

[00:05:55]You can connect me on telegram as well.

[00:05:57]You can connect we know we know the value of it is the same like this of x1 and x2 is equals to x1 + x2 upon 2 Add number divide by the number of three numbers x1 x2 x3 x1 + x2 + x3 upon 3 summation of xn upon on n where i = 1 to n summation of x i summation of x i upon n upon n that is i x1 + x2 + x3 upon 3 x1 + x2 + x4 upon 4 that is the definition of arithmetic Suppose if I asked you to find find arithmetic mean of 1 2 3 and four arithmetic Mean 1 2 3 4. What is the arithmetic mean of 1 2 3 4?

[00:07:52]Say what is arithmetic mean of 1 2 3 4?

[00:08:02]Arithmetic mean will be equals to 1 + 2 + 3 + 4 upon 4 that is 3 + 3 6 6 + 4 10 upon 4 that is equals to 5 upon 2 that is equals to 2.5.

[00:08:22]This is the arithmetic mean you will get the answer. This is the arithmetic mean of 1 2 3 4 3 581 3581 find 35811 3811 arithmetic mean 35811 And 358 11 27 upon 4 in RA 27 upon 4 perfect now let's come to the geometric mean that is arithmetic Now it's a condition Q should not be equals to Z condition should not be equals to Z right only positive number positive number positive Define of x1 x2 would be equals to x1 x2 the power 1 upon 2 geometric mean number x1 x2 x3 power 1 upon n 1 upon 3 1 upon 3 power 1 upon 2 1 upon roo<unk> -1 into possible root that would come in the imaginary geometric mean is defined only for positive numbers only for positive numbers root negative first.

[00:11:11]Find geometric mean of 2 and 4. Find geometric mean of 2 and four.

[00:11:29]Hi. Hi. Hello.

[00:11:33]Find geometric of 2 and 4 s 2 * 4 the power 1 upon n that is 1 upon 2 8 power 1 upon 2 that is roo<unk> 8 you can say or simplify a 2<unk>2 <unk>2 it's a generalization Solution 2<unk>2. All right. Very good. Neuron weight also good.

[00:12:09]All right.

[00:12:11]Good. Very good. Aa find geometric mean of find geometric mean of find geometric mean of 1 upon 8 Find geometric mean of 8 7 11 and 10. Find geometric mean of 8 7 11 and 10. Pometric mean of 8 7 11 and 10.

[00:13:04]Question mean 8 7 11 and 10 arithmetic mean of same number you will get a practice of it right simple Of course simple I also believe the same but for a practice point of view okay root 6160 root how it will come root generalization generalize x1 x2 up to xn power 1 upon n x1 x2 x up power up to X power upon N power upon N is generalize generaliz 8 7 11 and 10 8 7 11 and 10 8 7 11 10 the power 1 upon 4 power 1 upon 4 power 1 upon 4.

[00:14:45]All right, it's true. Now 6160 power 1 upon 4 it would come as 8.85 85 arithmetic mean check the lab bar very easy very easy arithmetic arithmetic mean very easy okay so let's come to harmonic mean what is progression is the next topic that is very important for inequalities Please question suppose hm of x1 and x2 am x1 x2 x1 + x2 upon 2 gm roo<unk> of x1 x2 or x1 x2 into x2 power 1 upon 2 hn 2 upon 1 upon x1 + 1 upon x2 2 upon 1 upon x1 + 1 upon x2 generalize n upon 1 upon x1 + 1 upon x2 up to 1 upon xn up to 1 upon xn digits number non upon x1 + 1 upon x2 suppose hm of three number h of three numbers 3 upon 1 upon x1 1 + 1on x2 + 1on x3 find harmonic harmonic mean of 3 56.

[00:16:47]Find harmonic mean of 3 five and six.

[00:16:51]Find harmonic mean of by way by find HM of 3 5 and 6. Find HM of 3 5 and 6.

[00:17:28]Say repeat.

[00:17:49]reciprocal that is 1 upon x1 + 1 upon x2 + 1 upon xn example 3 5 3 5 and 6 are three numbers right 1 upon 3 reciprocal sum 1 upon 5 + 1 upon 6 now How to solve this? You have to take the LCM then do the sum.

[00:18:16]You have to take the LCM then do the sum.

[00:18:21]Is it okay now? Cha.

[00:18:26]Is it okay now?

[00:18:30]Wait said 4.28.

[00:18:33]Check. Is the weight true?

[00:18:39]Is weight sing true?

[00:18:43]30 upon 7. Let me check now. What is of 35 and 6? Wait, what is the LC of 3, five, and 6?

[00:18:53]What is the LCM of 3, five, and 6?

[00:19:01]30. 30 is the LCM.

[00:19:04]10 + 6 + 5 that would come as 3 into 30 upon 21.

[00:19:14]3 upon 20 30 upon 21 right? That would come as 30 upon 7 with width is right.

[00:19:28]Uh no it's 30 upon 7 beta kani also right but gave the quickest answer that is fastest finger first all right it's 30 upon 7 upon A per for any given positive number. For any given positive number, Hm. Say am is always greater than equals to GM.

[00:20:28]Always equals to HM to prove this condition.

[00:20:43]Find AM GM HM of 2 and 3.

[00:21:03]you.

[00:21:11]Final.

[00:21:19]Who is going to support this team?

[00:21:22]GT versus RCP.

[00:21:24]Why? I will support RCB tomorrow.

[00:21:28]I will support RCB.

[00:21:45]Where upon 5 upon 2 GM is root 6. What is HN?

[00:21:55]What is HN?

[00:22:06]value of root 6 different that is 2.444 2.44 44 it is 2.5 what is HM 12.5 that is close to 2.4 2.4 2.44 44 what is the roo<unk> 6 accurate but am then comes GM then HM 2 value when you are going to take the SMS took the proof then it is HM Then comes the GM then comes.

[00:23:17]All right.

[00:23:18]All right. Perfect RCB or GD. Who's going to win tomorrow?

[00:23:43]AM= to GM. If all the terms are equal, find AM and GM of 333.

[00:24:00]Okay. RCB neuron. I don't watch cricket.

[00:24:04]Where you are from? Which state, country, city?

[00:24:09]Might be you might be from Dubai. It's all possible. I don't know.

[00:24:23]Hello.

[00:24:32]Okay. Also, don't watch.

[00:24:57]then AM is always greater than GM. If not then arithmetic mean it is coming same. Okay. RCB RCB open in the chat guys. RV X + 1 X= 2 X + 1 upon X= 2 X + 1 upon X would always be greater than 2 X + 1 upon X will always greater than 2. X + 1 upon X will always greater than 2.

[00:25:53]A m always greater than equals to gm x upon x value x or 1 upon x + 1 upon x upon 2 would be greater than =<unk> of x into 1 upon x upon 2 greater than equals to cut Cut up one.

[00:26:22]X + 1 upon X is always greater than equals to 2.

[00:26:45]With how many times you will say byebye greater than zero or if x is less than x less than it's less than equals tous 2 where to GM= This is a proof always mon x + 1 upon x upon That would be greater than equals to roo<unk> of x into 1 upon x= if minimum value of sum or maximum value of product is value.

[00:28:28]Max second terms involved in expression are true. Terms involved in expression are true. involved init good.

[00:29:09]Suppose m_sub_1 M1 or 1 upon M1 M1 or M1 cut a into a into 1 square A into A into 1 upon a square A into a A square. These are terms that are good and we can involve AM and GM concept. Aon B upon C upon A B say B C Example all variables should be positive or of same sign.

[00:30:09]Now let's come to the question part.

[00:30:18]Find the minimum value of x upon y + y upon x + yon + z upon y + x upon z upon x where xyz are all positive.

[00:30:36]May find the minimum value of this.

[00:30:48]Oh my m greater than equals to gm greater than equals to gm greater than equals to gm charge am greater than equals to gm greater than equals to gm M x upon y + y upon x + y upon z + z upon y + x upon z + z upon x upon ch should be greater than equals to should be greater than equals to x upon y into y upon X into Y upon Z into Z upon Y into X upon Z into Z upon X power 1 upon 6 power 1 upon 6 should be greater equals to GM Yon product of terms invol expression are good X Y Z X Z 1 1 power 1 upon That is one should be greater than equals to 6.

[00:32:52]Should be greater than equals to 6.

[00:32:57]Should be greater than equals to 6.

[00:33:15]Hello.

[00:33:19]Clear shi where I are from you. Where you are from?

[00:33:24]Where you are from? Which city? Which country?

[00:33:36]Which city? Which country?

[00:33:39]Which state?

[00:33:42]Chanka write chunka in the chat box. Shanka lanka sir. Shanka chamat West Bengal.

[00:33:56]Okay.

[00:34:05]Uh let me write this question.

[00:34:18]Yeah. Question.

[00:34:20]Find maximum value of Yeah. Question may 5 - x - 1 upon 2x maximum value 5 - x can you please repeat all right this is ation should be greater than equals to GM should be greater than equals to GM should be greater than equals to GM 1 2 3 4 5 6 that is terms sum upon should be greater equals to GM GMI* 1 that is 1on Multiply y X 1 minimum value expression greater than equals to 6 value six.

[00:36:00]Uh let me write again. Find minimum value of find minimum value of x + y + z * 1 upon x + 1 upon y + 1 upon z.

[00:36:29]Find minimum value of x + y + z into 1 upon x + 1 upon y + 1 upon z reciprocal reciprocal is that hm Mic greater than equals to GM greater than equals to HM GM HM greater than equals to HM hm answer.

[00:37:36]All right, everyone. It's appreciable guys. You are very active in the chat.

[00:37:40]I'm hoping in the notebook since I cannot see you.

[00:37:45]But I can hope you are solving the notebook as well.

[00:37:50]A h x + y + 1 upon x + 1 upon y + 1 upon x + y + x.

[00:38:14]You have learned the amm application guys application how you have to mold the concept in the defined format is x + y + z upon 3 that would be greater than equals reciprocal that would be 3 upon reciprocal 1 upon x + 1 upon y + 1 upon z greater than hm yes share session Those who are very active for the maths x + y + z into 1 upon x + 1 upon y + 1 upon z should be greater than equ= to 9 should be greater than equals to 9. Minimum value of x + yonals minimum value.

[00:39:54]No that is 9.

[00:39:59]But if you understood write chunka write chunka in the chat box guys. Write cha cha.

[00:40:14]Write chunka in the chat box guys. Write cha concept.

[00:40:23]How one beta? How one?

[00:40:47]N right?

[00:40:50]Is it is the concept clear?

[00:41:01]XY XY Z XY reciprocal.

[00:41:45]This method won't work. You have done.

[00:41:47]Okay, that is good. That is not an issue.

[00:42:17]If a and be a real number I guess it's not very clear if A and P are positive positive not positive right it's positive real numbers then now such that such that question such that such that a² + b² = Then find maximum value of AB.

[00:43:05]Then find then find maximum value of AB. Then find maximum value of AB.

[00:43:25]Understood.

[00:43:28]Yon X upon Y 1. I understood a boss such that a square b= 1. Then we have to find the maximum of a square or b square a square or b square.

[00:44:14]A m greater than equals to gm.

[00:44:18]A m greater than equals to gm. Okay.

[00:44:22]A square + B square A square + B square upon 2 A square B square A square + B square a square + B square should be greater than equ= to should be greater than equ= to should be greater than equals to roo<unk> of a² into b² under root of a square into b square.

[00:44:55]Find maximum value. Find maximum value of ab square root a square a that is modulus a modulus a positive b square modulus b or b positive a value. It is given a square + b square 1 to a square 1 that is 1 upon 2 greater than = a into b a square a b square b that would be it would be a and b such that therefore max value a is less than equals to 1 upon less Maximum it can be one upon two max one upon two is clear= 1 max 1 upon 2 maximum A key 1 upon 2. Max value A key 1 upon 2.

[00:46:38]Max value AB 1 upon 2. All right. Is it clear? Concept.

[00:46:46]Concept.

[00:46:52]concept.

[00:47:00]A b positive inteious real numbers such that= 1= 1 c should be greater than 8 1 upon a 1 - b 1 - c you prove Sir y 1 upon 2. The second part a square b square a square + b square upon 2 should be greater than equals to under root of a square into b square.

[00:47:58]Since a modal 1 upon 2 greater than equals to A= 1 Thank you.

[00:48:54]Hint a 1 - b minus c a 1 - b minus c2.

[00:49:19]So tell us a hint.

[00:49:50]It's awesome guys. You are almost we're going to end the session but you are live for the entire session.

[00:49:58]Good.

[00:50:04]Yes, you are in which class a= 1 - bon 1 + a 1 + a = 1 + 1 - b - c 1 + a = 1 - b + 1 - or gm.

[00:51:16]AMG upon two should be greater than equals to should be greater than equals to 1 upon B 1 upon C always= to GM 1 - B + 1 - C should be greater equ= to 2 into 1 - B 1 - CI are you here are you here in the chat Have a should be greater. 2 into 1 - b 1 - c 1 + a should be greater = 2 into 1 - b 1 - c how we went from this to this 1 + a greater = 2 under root 1 - b 1 - c 1 - b 1 - c Yeah.

[00:53:31]case. No, you just write b = 1 - a - c. Same procedure. 1 + b 1 + 1 - a - c is 1 1 - a + 1 - c is 1 - a + 1 upon c same procedure like this a concept 2 under root 1 - a 1 - c 2 under root 1 - a 1 - b multiply okay 1 + a 1 + b 1 + should be greater than equals to 2 into 2 into 2 that is 8 1 - a 1 - b 1 - cir great absolutely session, how the question asked, how the concept is molded. Mold ask questions.

[00:55:03]= 1 - c a 1= 1 - b + 1 - upon 2 should be greater than equals to roo<unk> of 1 - 1 - cm first step 1 - a minus Same.

[00:56:00]Same.

[00:56:08]Great. Great.

[00:56:10]Great.

[00:56:26]Find the maximum value of 5 - x - 1 upon x home work.

[00:56:35]Answer in the comment box is comment telegram discussion group as we are going to discuss this question on Monday session.

[00:56:47]Is it clear?

[00:56:49]Find the max value of xm.

[00:57:07]Thank you. Thank you so much.

[00:57:10]Thank you so much.

[00:57:15]upon discuss.

[00:57:25]All right. So we are ending session discussion.

[00:57:32]Second Google we will get your data. We will prepare accordingly for the live.

[00:57:44]Everyone, bye-bye. Good night. Shabbak. Sh.