Total Gadha-Maxima and MinimaFull description...

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Maxima, Minima Minima and Ine qualitiesqualities- The Basics by Total Gadha - Monday, 20 October 2008, 10:40 AM

This is a month of distress; students students going into into depression ove r their marks, eve ryone asking for attention, attention, frantic frantic phone calls and e mails, long hours of works hops, motivating motivating speeches, e xclusive sessions. In short, bullets flying all around and eve ryday bec oming a war zone. Par t of the game though. This is the month every instructor instructor in the field tightens his belt and gets ready for the barrage of querie s and emotions flying his way. (And I just burnt my tea I left on the burner 10 minutes ago while writing this. Oh well!) well!) I am still amazed how crushing those meaningless percentiles can be to the spirits of the students. students. I ke ep on telling students students don't take your perc entiles seriously. Don't take your percentiles seriously but my e xhortations xhortations always fall on deaf ears . Students Students are so much caught in this web that they they cannot detect that that half their miseries are emanating from something that is not real and cannot supplant the the re al thing- The CAT. O h well, I better better go and answer those distress calls. For all those s tudents tudents telling me that I have disappeared from TG, he re is the new chapter to shush them for a while. Till the mutiny rises again

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I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this in the CBT Club this week.

If you think this article was useful, help others by sharing it with your friends!

You might a lso like: Quadratic Equation Absolute Value (Modulus) Function totalgadha.com/mod/forum/discuss.php?d=4040

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Re: Maxima, Minima and I nequalities- The Basics by vamsi krishna - Monda y, 20 October 2008, 10:55 AM Oh mY...... Inequalities Simplified.... SIR,,,we de mand s ome exercises to munch on VaMsI Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Kitty Witty - Monday, 20 October 2008, 06:12 PM

Sir Hats off to you for such an ins ightful article. Could you plz help me solve this problem using AM > GM funda.... Let x,y, z be distinct positive integers such that x+y +z=11. Find the maximum value of (xyz+x y+yz+zx )? This is how i approached the problem. for product of any 2 nos to be maximum they shld be as close to each other as possible...... using this x=2,y=4,z=5 substituting (xyz+xy+yz+zx) =78 However using AM > GM funda is more fool proof.... can some one help me with that? Regards, Kitty Witty Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by ATOM ANT - Monday, 20 October 2008, 06:52 PM Thanks for the lesson sir.. Can you explain how you found the base and height of the triangle in the second exa mple ...

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Re: Maxima, Minima and I nequalities- The Basics by Total Gadha - Monday, 20 October 2008, 07:14 PM Hi Kitty, take 4, 4, 3. The numbers should be nearly equal. Maximum value = 88.

Total Gadha Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by rashi agarwal - Monday, 20 October 2008, 07:51 PM

thankyou TG s ir for such a wonderful article.We are really in need of m ore such articles. sir I have a doubt in this ques. find the minimum value of |x-1|+ |x-3| +|x-10|. In this, can we take any two values a s a a nd b so that |x-a| +|x-b| remain constant .I ha vnt got the solution of this one.why have we take n a and b as 1 and 10? wht not 3 and 10? there is one m ore question of the sam e type. i havent got that also.please explain this question. regards rashi Show parent | Reply

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Re: Maxima, Minima and I nequalities- The Basics by whirl wind - Tuesday, 21 October 2008, 01:28 AM

TG, Glad to see that. If only all that u need is a m utiny for ur presence here - why - u can see o ne any time...and i dont mind starting one right now - and i don t think others will be late in joining me in the mutiny for an article of urs. U've unanim ousy estab lished the unm atched quality of content o n TG m any times befo re...we ne ed nt te ll wat the content o f the ne xt article sh d be - u k now it much better. But we will be waiting for one.

Btw, TG, wat abo ut the solutions of CC -4 and CC-6??Ha vent had them yet - desp erately waiting for them.. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Kitty Witty - Tuesday, 21 October 2008, 06:18 AM

Hi TG Sir, Thank s for ur quick reply. However the ques tion states that x ,y and z a re distinct integers. so 4,4,3 will be ruled out. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Kitty Witty - Tuesday, 21 October 2008, 07:13 AM

Another Problem Sourced from Quant Marathon Blog 1)Two real non negative numbers satisfy that ab>=a^3+b^3, f ind the maximum value of a+b a) 1/2 b) 1 c) 3/2 d) 2 e) none of these My Approach a^3+b^3 >= ab(a+b) if a>0,b>0 .......... ...... .....1 This implies.... the least value possible of a3+b3 = ab(a+b) a^3+b^3 <= ab (problem stmt)......... ...... ...... ..2 ab(a+b) <= ab........... ...... ..from 1 and 2 possible only if a+b =1 Not sure if a nswer is correct....approach also see ms very crude...

TG Sir and junta ...pour in your approaches.......

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Re: Maxima, Minima and I nequalities- The Basics by Chinmay Korhalkar - Tuesday, 21 October 2008, 09:37 AM Hi Sir, Thank s fo r such a wonderful article. I've 2 questions from last year's AIMCAT. There a re three natural numbe rs x,y & z such that 2x+3y+4z = 100. 1. What is the maximum value of x^2+y62+z^2? a)2500 b)2041 c)2036 d)2030 e)2024 2. What is the maximum value of 2x^2+5y^2+8z^2? a)4066 b)4526 c)4534 d)4672 e)4754

Regards. chinmay.

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Re: Maxima, Minima and I nequalities- The Basics by zico on the run - Tuesday, 21 October 2008, 07:26 PM

Hi Tg , wante d to clarify whethe r the " the condition for ax 2 + bx + c>=0 is a>0 and b 2-4ac<=0" means that when ax 2+bx+c is always lesse r than zero implies a>0 and b 2-4a c <=0. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by priyanka tiwari - Wednesday, 22 October 2008, 09:03 AM

in ex 7 why has the condition of imaginary roots being used? plz explain Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Venkkatesan R - Wednesday, 22 October 2008, 09:11 AM Solve for the intersection o f two eq uation, you will get the heig ht of the triangle. Substitute y=0 u will get a value of x for each equa tion. The differnce of these two values will give u the ba se... Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Venkkatesan R - Wednesday, 22 October 2008, 09:20 AM

1) x=45,y=2,z=1 and a ns= 2030??? Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by saurabh k - Wednesday, 22 October 2008, 02:55 PM

Wow Awesome Article Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by saurabh k - Wednesday, 22 October 2008, 03:43 PM

One query junta In case of problems invovling maximum and minimum values of a symmetrical expression, we put equal values of variables involved to find max or min. But is it possible to distingush b/w max and min? Is it really possible to find the max and min both, given sum of variables invovled for a given symmetrical expression. x, y,z non -ve real numbe rs. x+ y+ z = 1. Find ma x and min for

x/(1+yz) + y/(1+xz) + z/(1+xy).

I can find 9/10 by putting x =y=z=1/3. Now how to find the other value invovled? Junta ple ase reply. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Rishi Kapoor - Wednesday, 22 October 2008, 04:26 PM Chinmay, Solutions Plea se! Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by ATOM ANT - Wednesday, 22 October 2008, 06:34 PM Thanks Venkatesh. Show parent | Reply

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Re: Maxima, Minima and I nequalities- The Basics by ashish sharma - Wednesday, 22 October 2008, 09:26 PM Hi...TG Sir....in the era of commercialisation of education and exploitation of money...some very noble and generous concepts come from your side. Sir i have joined a coaching center, paid a hefty amount and taught by many teachers but honestly the concepts you give are so grasping and useful that i have found them m uch more useful than any material.I have m ade up m y mind if in this year i dont get selected i will join your coaching .... Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Rishi Kapoor - Wednesday, 22 October 2008, 09:34 PM How do we came to know which one we have to ma ke the greatest? Please Reply Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Nikhil Dhand a - Thursday, 23 October 2008, 11:52 AM hey my ap proach for this was : given : ab >= a^3+b^3 but we k now that (a ^3+b^3)/2 > ((a+b)/2)^3 from theorem given a bove hence ab >= ((a+b)/2)^3 but we know that AP>=GP ie (a +b)/2 >= (ab)^1/2 taking squares ((a+b)/2)^2 >= ab >= ((a+b)/2)^3 hence a+b Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Nikhil Dhand a - Thursday, 23 October 2008, 03:24 PM Whats the approach for this qs??...people who got the answer plz give ur approach for the same Thnks Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by dhwani parikh - Sunday, 9 November 2008, 09:47 PM just superb..it helped me a lot in thes e las t days... and the b est thing is chess picture .... u've related it with this wonde rfully.. great on e... Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by nitesh a garwal - Saturday, 15 Novemb er 2008, 03:12 PM the other value is 1 when x=y=0 and z=1 Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics

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MBA|CAT|CAT 2011|CAT 2012|CAT Online|MBA 2012|MBA Entrance Exams|CAT Test|Preparation|CAT Questions by shivan i tiwari - Thursday, 27 Novembe r 2008, 11:41 AM

Although I am not so regular on TG but whenevr I ge t a chance to log on I just find a single word "MIND BLOWING". Last time I ge t to know how to find the la st two digits o f a num ber raised to a ny power ..that was yet an other fantabulous article by TG so lucid ..the sam e is with this one .. Rea lly you are simply mind blowing in your way of e xplaining..

Thanks for all your efforts..hope to find some more articles on these topics.. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by amit ranjan - Thursday, 30 July 2009, 07:38 PM Hi TG, Thanks a lot TotalGadha bhai. Very very useful stuff. Best Regards, Amit Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Nitin Kumar - Saturday, 5 September 2009, 01:09 PM we need to take distinct postive integers. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Arundeep Raina - Wednesda y, 16 September 2009, 02:45 PM Plz explain if there is any other approach to be used in such questions ( apart from making one number greatest and the others as least as possible). can anybody explain the solution for 2nd ques? Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Ankit Talwar - Monday, 26 October 2009, 03:06 PM

Hi all, The appr oach to solve the above mentioned pr oblems is to maximize one of the number s while minimizing the other two. Ans 1: Take x=50, y=0,z=0; the maximum value of ( x^2 + y^2 + z^2) = 2500. Ans 2: We have to see the weights attached with dif ferent numbers. In the second p r oblem. z has the highest weight attached to it { 8(W z) > 5(Wy) >2(Wx)}. So we will take the following case x=0, y=0, z =25; the maximum value of (2x^2+ 5y^2 + 8z^2) = 5000 (which is not p r esent as any of the options.) Chinmay I would r equest you to check the answer for the second and pr ovide clarifications if r equired. Regards

Ankit Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by shravan kuma r - Sunday, 8 November 2009, 04:43 PM

@ Ankit.. ..Dude...please redo your calculations cos x, y, z are natural numbers. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by nishchai nevrekar - Sunday, 8 November 2009, 06:42 PM ne one with viable solns to these problems.... so tht thr is some generalized method which can be xtended to other problems like these.... @chinmay. .... dude can u post those aimcat soln if possible... Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics

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MBA|CAT|CAT 2011|CAT 2012|CAT Online|MBA 2012|MBA Entrance Exams|CAT Test|Preparation|CAT Questions by nidhi soni - Saturday, 14 Novem ber 2009, 10:20 PM

really really very gud chp tg thnku so much its really helpful Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by payal saraf - Sunday, 29 November 2009, 05:14 PM

plz help x/(x 2 -5x +9)

in d ques.(e g. 6) " find max n min value f function y=

after d step (11y+1)(y-1) less than e qual to 0, we can hv 2 cases 1. 11y+1 less than equal to 0 and y-1 greater than equal to 0 therefore, y less than equal to -1/11 and y greater than equal to 1 or 2. 11y+1 greater than equal to 0 and y-1 less than equal to 0 therefore -1/11 (less than = to) y (less than = to ) 1 then why do we choos e case 2 only?????

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Re: Maxima, Minima and I nequalities- The Basics by Subhash Medhi - Tuesday, 1 June 2010, 02:39 AM Dear TG sir, I too have the same doubt as Saurabh K. Is it poss ible in case of symmetrical expressions to distinguish between m aximum and minimum values. In the article, it is given that to find maximum or m inimum values in case of symmetrical expressions we have to assign equal values to each of the variables.Does it mean that in case of symmetrical expressions max imum and minimum values are equal ? C an that really be the case ? Or should we assign equal values to the variables only while calculating the minimum value? Regards, Subhash Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Pravin Vaidya - Wednesday, 2 June 2010, 12:36 AM

simple doubt?? solve the inequality... (X/4) + (2/3) < (2X/3)-(1/6) ==> 1st approach (X/4)-(2X/3) < -(1/6) -(2/3) -5X/12 < -(5/6)===> (X/12) < (1/6) which gives, X<2 2nd approach (2/3)+(1/6) < (2X/3)-(X/4) (5/6) <5X/12 which gives, X>2 I am getting two diffe rent solution ,could you plea se tell m e which one is right?? and why the other one is wrong. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by TG Team - Wednesday, 2 June 2010, 11:49 AM

Pravin totalgadha.com/mod/forum/discuss.php?d=4040

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See -5 < -2 does that mean 5 < 2. I hope you can find your error now. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Pravin Vaidya - Thursday, 3 June 2010, 01:39 PM

Thanks kam al.... Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Naman Mirchandani - Saturday, 19 June 2010, 11:25 AM

Sir, For the 2nd la st ques tion where a+b+c = 1, how will we kno w that p utting a =b=c, will give the mxm or minm value of (1/a - 1) (1/b - 1) (1/c - 1) ?

Thanks,

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Re: Maxima, Minima and I nequalities- The Basics by Ramak anth Kanagovi - Monday, 12 July 2010, 04:30 PM

1)Two real non negative numbers satisfy that ab>=a^3+b^3, f ind the maximum value of a+b a) 1/2 b) 1 c) 3/2 d) 2 e) none of these i think the answer is 1/2 taking the values as 1/4 and 1/4 Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Abhirup DebRay - Monday, 12 July 2010, 07:34 PM thnk option b 4 a=b=1/2 Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Navneet H - Wednesday, 22 December 2010, 07:45 PM

Hi kitty, The values that you have taken are not close to each other.The value of x y and z should be 11/3.. in that case the result of te expression will be: 242 9/27 which is slightly greater than 89.666

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Re: Maxima, Minima and I nequalities- The Basics by rajesh mishra - Saturday, 30 July 2011, 01:00 AM what in case if the mod value changes like |1-x|+|2-x|+|3.5-x|+|x-4| can u please explain this in terms of distances as explained above? thanks Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by manisha dalan - Tuesday, 2 August 2011, 04:55 PM

dea r sir,

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MBA|CAT|CAT 2011|CAT 2012|CAT Online|MBA 2012|MBA Entrance Exams|CAT Test|Preparation|CAT Questions I a m really grateful to you for this wonderful article. Regards, manisha. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by TG Team - Wednesday, 3 August 2011, 01:46 PM

Hi Rajesh Understand the concept clearly. On number line '5' denotes a point which is 5 unit away from origin on the right side. Right? Similarly 'x' denotes a point which lies at a distance of 'x' units from origin. So |x - 5| denotes the distance between two points 'x' and '5' on the number line. What does |5 - x| represent on number line? Isn't it same the distance between two points '5' and 'x' on the number line? It is. Hope it is clear. Kamal Lohia Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by anupam chaturvedi - Thursday, 25 August 2011, 03:33 PM Hi TG Sir, I am still not able to understand the concept of symmetrical expressions. What does it exactly mean for below questions! 1) --> Min of (a1+a2+a3+a4)(1/a1 + 1/a2 + 1/a3 + 1/a4) I solved it using max product rule. How to a pply symm etry here? --> a+b+c=18 and we ne ed to find MI N of (1/a -1)(1/b -1)(1/c -1) Is (a +b+c) called s ymme trical OR (1/a -1)(1/b -1)(1/c -1) ? 2) (a1+a2+a3+a4)(1/a1 + 1/a2 + 1/a3 + 1/a4) how can we find the ma x. value o f abo ve function..?? Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by abhinay dutta - Thursday, 8 September 2011, 11:26 AM

Sir Hats off to you for such an ins ightful article. Could you plz help me solve this problem using AM > GM funda.... Let x,y, z be distinct positive integers such that x+y +z=11. Find the maximum value of (xyz+x y+yz+zx )? This is how i approached the problem. for product of any 2 nos to be maximum they shld be as close to each other as possible...... using this x=2,y=4,z=5 substituting (xyz+xy+yz+zx) =78

had u calculated values for x,y,z as 4,4,3 answer wud hv been 88 more than 78 and is Maximum. Guys can u post more question on Max and min or can u site some sources for practise. Thanks

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Hi Abhinay xyz + xy + yz + zx = (x + 1)(y + 1)(z + 1) - (x + y + z) - 1 And this will be maximum when (x + 1)(y + 1)(z + 1) is maximum. We know that sum of these three terms (i.e. x + 1 + y + 1 + z + 1 = 14) is constant, so there product will be maximum when these three terms are as close as possible.(preferably equal) But in this question, it is given that x, y, z are distinct positive integers so (x + 1), (y + 1), (z + 1) should also be distinct but should be close also. So the optimum case is 6, 5, 3 and the required maximum product will be 6 × 5 × 3 11 - 1 = 78. Kamal Lohia Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by priyanka j - Thursday, 8 September 2011, 02:08 PM

f(x)=min(5-x,x+3), find max value of f(x) Sir, In these type of que stions is it ne cessary to draw graph can't we directly find out the intersection of b oth lines b y directly putting them equa l to each other. like in this case 5-x=x+3 gives x=1 . max value of f(x)=4 at x=1 Thanks Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by TG Team - Thursday, 8 September 2011, 02:38 PM

Priyanka In this case, it's ok to just equate the expressions and get the value of x at which f(x) attains its maximum. But if f(x) = min(5 + 3x, x + 3), then what will you do? Just try and think. Kamal Lohia Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by priyanka j - Thursday, 8 September 2011, 03:25 PM Thanks sir. it will nt work in evry case . but in d ques u have written is it possible to find any mix max or min value. bcz d value ranges frm -infinity to +infinity. As value of f(x) is increasing as x increases & decreasing as x decreases. What i think is there must be any limit for value of x, then only it is possible to find min or max value of f(x). I know m asking silly things but plz clear my doubts. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Jitendra Soni - Friday, 9 September 2011, 03:18 PM My solutions: A) using x ^2 + y^2 + z^2 >= ((x+y+z)^2)/3 ..... (i) , we ge t from the give n conditio n 2x+3y+4z=100 2(x+y+z) + y+2z=100 or x+y+z= 50-(y+2z)/2 ..now y + 2z sho uld be divisible by 2. As all x,y,z a re natural nos. h ence for m ax value of x + y + z, we need min value of y+2z (which should be even also to give integral value of x+y+z.Trying with y+2z =4 , we get y=2 and z=1 (with 2 we will not get natural nos.) hence x+y+z = 48. Now substitute in x^2 + y^2 + z^2 to get the ans B) se parate all terms , we g et x ^2+ x^2+ y^2+ y^2+ y^2+ y^2+ y^2+ z^2+ z^2+ z^2 +z^2+ z^2+ z^2+ z^2+ z^2 >= ((x+x+y+y+y+y+y+z+z+z+z+z+z+z+z)^2)/15 = ((2x+5y+8z)^2)/15..= ((200-(2x+y))^2)/15 , now we have to minimize 2x+y subject to the condition that z = ( 25-(2x+3y)/4) will be an integer. This gives 2x +3y=8 ( = 4 does no t give natural no. sol.) he nce x= 1, y= 2 and z= 23. Substitution in 2x ^2+5y^2+8z^2 g ives the ans. Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by sandesh gupta - Monday, 12 September 2011, 04:47 PM

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MBA|CAT|CAT 2011|CAT 2012|CAT Online|MBA 2012|MBA Entrance Exams|CAT Test|Preparation|CAT Questions Thanks a lot sir for such a wonderful article . Sir can you please explain how yo solve question like |X -3| + |Y - 4| = 5 (for ex) kind of question . I am always stuck in this kind of problem . Thanks & Rega rds sandesh(coming out from my IT background

) Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by Rahul Sharma - Wednesday, 14 September 2011, 11:33 AM Hi sir, min value of |x-1|+|x-10| as per my understanding, the value of the expression will be a constant when x lies between max and the min values(i.e 1, 10). when there are other terms included in between then the value of the expression will largely depend on them.if we have odd terms in all, then the value of the expression will b max at the value of x obtained from the middle term. If the no of terms are even the the two middle terms will decide the range of x for the expression to have max value.

please let me know if i m right. And alsoe please help me in finding the values of the following expressions: |1-x|+|x-3|+|x-10| |x^2-1|+|x2-3|+|x^2-10| Show parent | Reply

Inequalities concepts by Dinesh H - Thursday, 21 Feb ruary 2013, 12:00 AM For Inequalities concepts pls visit http://start-from-scratch-cat.blogspot.in/2013/02/cat-inequalities-concepts.html Show parent | Reply

Re: Maxima, Minima and I nequalities- The Basics by ashwini rathore - Friday, 22 February 2013, 11:41 PM hi this is yash... i m going to tell my method F(X)=|X-a|+|X-b|+|X-c| min value always o ccur At b if a

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