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Suppose that |x + y| + |x -y| = 2. What is the maximum possible value

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Suppose that |x + y| + |x -y| = 2. What is the maximum possible value  [#permalink]

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New post 31 Mar 2019, 22:15
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Question Stats:

62% (02:30) correct 38% (02:20) wrong based on 117 sessions

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Re: Suppose that |x + y| + |x -y| = 2. What is the maximum possible value  [#permalink]

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New post 31 Mar 2019, 22:46
2
if we open mod there are 4 cases possible

Case 1) Both are positive

x + y + x - y = 2
2x = 2
x = 1

Case 2) |x + y| is negative

-x - y + x - y = 2
-2y = 2
y = -1

Case 3) |x - y| is negative
x + y - x + y = 2
2y = 2
y = 1

Case 4) Both are negative
- x - y - x + y = 2
-2x = 2
x = -1

So from the above 4 cases, we can say \(x = \pm{1}\) and \(y = \pm{1}\)

Substituting in the eq: \(x^2 - 6x + y^2\)

\(1 - 6(\pm{1}) + 1\)

This value will be maximum when x = -1

=\(1 -6(-1) + 1\)

=8

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Re: Suppose that |x + y| + |x -y| = 2. What is the maximum possible value  [#permalink]

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New post 01 Apr 2019, 00:25
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Bunuel wrote:
Suppose that |x + y| + |x -y| = 2. What is the maximum possible value of x^2 - 6x + y^2?

A. 5
B. 6
C. 7
D. 8
E. 9


We can find our solution by logic.
(I) Both the terms |x + y| and |x -y| are at least 0, and cannot be more than 2 as their sum itself is given as 2.
(II) If you have to maximize \(x^2 - 6x + y^2\), we will try to maximize -6x as that will be the largest value, so x should be negative.

Now |x| and |y| should not be greater than 1, so both can be 1..
x has to be negative, so x=-1, and y =1..
|x + y| + |x -y| =|-1+1|+|-1-1|= 2...Yes

Value of \(x^2 - 6x + y^2\) will become \((-1)^2 - 6(-1) + (-1)^2=1+6+1=8\)


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Re: Suppose that |x + y| + |x -y| = 2. What is the maximum possible value  [#permalink]

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New post 01 Apr 2019, 17:58
1
How I approached this question is as follows.
I know that every term in equation is positive. Therefore, I squared both sides.
And got (x^2 + y^2)= 2.

Now I have maximize the x^2 + y^2 - 6x
To maximize 2-6x, x must be negative and can be -1.
So, maximum value is 8.

I think my approach to such questions in which both sides are positive is correct.

chetan2u wrote:
Bunuel wrote:
Suppose that |x + y| + |x -y| = 2. What is the maximum possible value of x^2 - 6x + y^2?

A. 5
B. 6
C. 7
D. 8
E. 9


We can find our solution by logic.
(I) Both the terms |x + y| and |x -y| are at least 0, and cannot be more than 2 as their sum itself is given as 2.
(II) If you have to maximize \(x^2 - 6x + y^2\), we will try to maximize -6x as that will be the largest value, so x should be negative.

Now |x| and |y| should not be greater than 1, so both can be 1..
x has to be negative, so x=-1, and y =1..
|x + y| + |x -y| =|-1+1|+|-1-1|= 2...Yes

Value of \(x^2 - 6x + y^2\) will become \((-1)^2 - 6(-1) + (-1)^2=1+6+1=8\)


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Re: Suppose that |x + y| + |x -y| = 2. What is the maximum possible value  [#permalink]

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New post 04 Apr 2019, 09:38
1
gvij2017 wrote:
How I approached this question is as follows.
I know that every term in equation is positive. Therefore, I squared both sides.
And got (x^2 + y^2)= 2.

Now I have maximize the x^2 + y^2 - 6x
To maximize 2-6x, x must be negative and can be -1.
So, maximum value is 8.

I think my approach to such questions in which both sides are positive is correct.

chetan2u wrote:

Hi , I'm curious , when you square both sides , you got 2 *|x + y| * |x -y| in your way , how you did deal with that?
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Suppose that |x + y| + |x − y| = 2. What is the maximum possible value  [#permalink]

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New post 10 May 2019, 22:43
1
(|x+y| + |x-y|) ^2 = 4
2(x^2 + y^2) + 2( x^2-y^2)=4
4x^2=4
x=+/- 1

to maximize, x must be -1

put this in original equation, y can be 0 or 1

since we are looking to maximize, choose 1

Answer= 1+6+1= 8
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Suppose that |x + y| + |x − y| = 2. What is the maximum possible value   [#permalink] 10 May 2019, 22:43
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