New Batches at TathaGat Delhi & Noida!               Directions to CP centre
Maximum / Minimum type of questions
by the underdog - Monday, 6 August 2007, 04:06 PM
  (Q) If a, b, c are positive real numbers such that a2 + b2 + c2 = 48, then the maximum value of a+b+c is
(1) 9 (2) 16 (3) 12 (4) 24

Can someone please advice as to how should one tackle such maximum / minimum related questions? How can one get good at this (obviously practise but there must be certain key approaches right?).

Thanks!
Re: Maximum / Minimum type of questions
by akash mittal - Monday, 6 August 2007, 05:19 PM
 

22hi underdog

since (a+b+c)2

=a2+b2+c2+2(ab+bc+ca)

=48+2(ab+bc+ca)

here u have to divide 48 into three parts of a2,b2,c2 that is into a,b,c such that their products ab,bc,ca are maximum.this can be done when all values are as close as possible.that is a2=b2=c2=48/3=16.that is a=b=c=4

then maximum a+b+c=4+4+4=12.

hope u understood the concept.any more queries with maximum/minimum?

 

Re: Maximum / Minimum type of questions
by the underdog - Monday, 6 August 2007, 05:46 PM
  Hey thanks! I got what you're saying, but I have one doubt.

Isn't there a rule which states that "if the product of two positive quantities is given, their sum is least when they are equal.".... but we are trying to maximize the sum here!

You have said:

here u have to divide 48 into three parts of a2,b2,c2 that is into a,b,c such that their products ab,bc,ca are maximum

So how come you came upon this conclusion? I mean how did you know that we have to divide 48 in such a way?


Re: Maximum / Minimum type of questions
by Sri KLR - Tuesday, 7 August 2007, 11:45 AM
 

The only thing I know about max/min is

i) if sum is given, product will be max when the numbers are equal.

ii) If the product is given, the sum will be min when the numbers are equal.

If anybody can give a convincing soln to htis...u are welcome( I think the ans is 2)

1. if xy + yz + zx = 2 whats the minimum value of x ^2 + y ^ 2 + z ^ 2?

Re: Maximum / Minimum type of questions
by the underdog - Tuesday, 7 August 2007, 12:13 PM
  Hi Sri,

The answer is option (3), but I'm not convinced in the sense that by ensuring products ab,bc,ca are maximum by ensuring the values to be as close as possible, we would result in the least sum (as you too have mentioned in rule (ii)), but we are trying to maximize the sum here. Waiting for akash's explanation.
Re: Maximum / Minimum type of questions
by the underdog - Tuesday, 7 August 2007, 12:22 PM
  (Q) If x3y4 = 36; then what is the least value of 4x+9y?

(a) 16 (b) 18 (c) 14 (d) 28 (e) 21
Re: Maximum / Minimum type of questions
by lets try - Tuesday, 7 August 2007, 12:43 PM
  Hi underdog!!

reference:
(Q) If x^3 * y^4 = 36; then what is the least value of 4x+9y?

take 4x = 3X
and 9y = 4Y
easy to observe ..why?
putting it in
x^3 * y^4 = 36
we get X^3 * Y^4 = 3^7
so the sum 3X + 4Y to be minimum X = Y = 3
so 4x + 9y = 3X + 4Y = 21

Thanks,
Jitendra.
Re: Maximum / Minimum type of questions
by Sanju P - Tuesday, 7 August 2007, 01:04 PM
 

Hi Underdog,

What i understand is - If (a + b + c) { hereafter, will be referred as A } is max, then ( a + b + c)^2 is also maximum..

Here, since the sum of suares of a. b & c are given, we need to see the relation b/w the given value n the required one.

So,  A^ 2 = a ^2 + b^2 + c^2 + 2(ab + bc + ca)   

Let the 1st 3 terms in RHS be B.

Now, B is given to be 48.

for LHS ( A^2) to be max, only thing we can maximise would be the second term of RHS since B is constant ( = 48).

Now we have B = 48, so we'll get the max product if we have all its terms equal... rest follows..  

Thanks

SP smile

Re: Maximum / Minimum type of questions
by the underdog - Tuesday, 7 August 2007, 02:34 PM
  1. if xy + yz + zx = 2 whats the minimum value of x ^2 + y ^ 2 + z ^ 2?

Is the answer 2?
Re: Maximum / Minimum type of questions
by Kuntal Roy - Wednesday, 22 April 2015, 10:02 AM
  (b^2 + c^2) >= (b ^2 .c ^2) (using AM>=GM for positive real numbers  )
or , (b^2 + c^2) >= bc
similarly (a^2 + b^2) >= ab
(a^2 + c^2) >= ca

hence ab+bc+ca <= (a^2 +b^2 + c^2) = 48

again (a+b+c)^2= (a^2 +b^2 + c^2) + 2(ab+bc+ca)

(a+b+c)^2= 48 + 2(ab+bc+ca)

now (a+b+c)^2 is max when (ab+bc+ca) is max

hence maximum value of (a+b+c)^2  = 48+2x48= 144

again since a,b,c are real and positive a+b+c is max when (a+b+c)^2 is max

hence maximum value of (a+b+c) is = sqrt(144) =12