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Algebra

Scorching goIITian

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1 Nov 2008 21:36:46 IST

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786

plz solve this(CAT 2002) question

None

if x,y,z are real numbers and x+y+z=5 , xy+yz+xz=3 then the maximum value x can have is ?

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Mirka

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1 Nov 2008 22:01:27 IST

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Isn't this similar to this question (from IIT sample paper 2006)
Refer to this one :
http://www.goiit.com/posts/list/algebra-x-y-z-15-and-xy-yz-zx-72-prove-that-3-x-7-88587.htm

sriram

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2 Nov 2008 07:34:55 IST

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i think the answer is "3"

the procedure is

given..,

x+y+z=5

squaring on both sides

x²+y²+z²+2{xy+yz+xz}=25

x²+y²+z² +2{3}=25

x²+y²+z² =25-6

=19

the only triplet satifying this condition is 3,3,-1

bcozzzzz we can take -3,-3,1 also but their sum is not equal to 5

so the triplet is 3,3,-1 this satisfies all the given conditions so the max value for x is 3 and min value is -1

rate if useful!!!!!!!!!!

Ankush Jain

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2 Nov 2008 08:07:56 IST

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x+y+z=5

squaring on both sides

x²+y²+z² +2{3}=25

x²+y²+z² =25-6

=19

But now we know that

Hence

Hence max. value is 25/3

sourab bose

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2 Nov 2008 08:31:28 IST

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unfortunately, neither of the above answers is correct :(

santhosh in NUS

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2 Nov 2008 09:11:46 IST

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IS ans 13/3

Hari Shankar

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2 Nov 2008 09:45:50 IST

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From the given equalities, we get $x^2+y^2+z^2 = (x+y+z)^2-2\sum xy = 19$

Hence $y^2+z^2 = 19-x^2 \ge 2yz$

Now, $3 = xy+yz+zx = x(y+z)+yz = x(5-x)+yz \le 5x-x^2+\frac{19-x^2}{2}$

This leads to the inequality, $3x^2-10x-13 \le 0$

and since the leading coefficient is positive, we have $x \le \frac{13}{3}$

Rahul Duggal

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2 Nov 2008 10:17:35 IST

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Re:plz solve this(CAT 2002) question

sourab bose

Scorching goIITian

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2 Nov 2008 14:18:12 IST

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yes , the answer is 13/3

sourab bose

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2 Nov 2008 14:24:00 IST

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i have a doubt regarding this question, is it true that for x to be maximum y must equal z i.e y=z.
if so, then how come it can be inferred?

Conjurer

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2 Nov 2008 14:35:01 IST

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Another

x^2 + y^2 + z^2 = 19

Using cauchy swarz

(y^2 + z^2)(1 + 1) > = (y+z)^2

Substituting to get x,

=> (19 - x^2)(2) >= (5-x)^2

=> 38 - 2x^2 > = 25 + x^2 -10x

=> 13 >= 3x^2 -10x

Conjurer

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2 Nov 2008 14:35:49 IST

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Equality holding for cauchy swarz when y/1 = z/1

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