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Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2

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Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2 [#permalink]

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New post 16 Mar 2017, 01:30
4
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

30% (01:01) correct 70% (01:25) wrong based on 63 sessions

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Is \(x^3+y^3>x^2 +y^2\)?

(1) \(x+y > x^2+y^2\)
(2) \(x^4 + y^4 > x^2+y^2\)

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Re: Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2 [#permalink]

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New post 16 Mar 2017, 05:02
Is x^3 + y^3 > x^2 + y^2 ?
x^3 - x^2 + y^3 - y^2 > 0?
x^2(x - 1) + y^2(y - 1) > 0?

St1: x + y > x^2 + y^2
x^2 - x + y^2 - y < 0
x(x - 1) + y(y - 1) < 0 --> Possible when x is between 0 and 1 and y is between 0 and 1.
Hence x^2(x - 1) + y^2(y - 1) is not greater than 0.
Sufficient.

St2: x^4 - x^2 +y^4 - y^2 > 0
x^2(x^2 - 1) + y^2(y^2 - 1) > 0 --> x and y can be -ve or +ve
If x < -1 and y < -1 then x^3 + y^3 cannot be greater than x^2 + y^2
If x > 1 and y > 1 then x^3 + y^3 > x^2 + y^2.
Not Sufficient.

I think the answer has to be A and not E. Let me know if I have made an error.
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Re: Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2 [#permalink]

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New post 17 Mar 2017, 10:14
Vyshak wrote:
Is x^3 + y^3 > x^2 + y^2 ?
x^3 - x^2 + y^3 - y^2 > 0?
x^2(x - 1) + y^2(y - 1) > 0?

St1: x + y > x^2 + y^2
x^2 - x + y^2 - y < 0
x(x - 1) + y(y - 1) < 0 --> Possible when x is between 0 and 1 and y is between 0 and 1.

x(x - 1) + y(y - 1) < 0 does not require that both x and y be between 0 and 1.
If x=0.2 and y=1.1, then x(x - 1) + y(y - 1) = (0.2)(-0.8) + (1.1)(0.1) = -0.16 + 0.11 = -0.05.

Quote:
Hence x^2(x - 1) + y^2(y - 1) is not greater than 0.

If x=0.2 and y=1.1. then x²(x - 1) + y²(y - 1) = (0.04)(-0.8) + (1.21)(0.1) = -0.032 + 0.121 = 0.089, which is greater than 0.

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Re: Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2 [#permalink]

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New post 17 Mar 2017, 19:45
ziyuen wrote:
Is \(x^3+y^3>x^2 +y^2\)?

(1) \(x+y > x^2+y^2\)
(2) \(x^4 + y^4 > x^2+y^2\)


The first statement implies 0<x<1 and 0<y<1.

And second implies x>0 and y>0.

So IMO answer should be D.

Where am I going wrong here? :?
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Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2 [#permalink]

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New post 18 Mar 2017, 08:03
1
Statement 1 is not sufficient because there can be two scenarios.

1. 0<X<1 and 0<Y<1. in this case x^3 + y^3 > x^2 + y^2 is not true

2. Either X or Y is between (0;1) and the other is slightly bigger than 1.
For example X=1.045 and Y=0.95
X+Y=1.995, X^2+Y^2=1.994525 and X^3+Y^3=1.998
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Is x^3+y^3>x^2 +y^2? (1)x+y>x^2+y^2 (2) x^4 + y^4 > x^2 + y^2   [#permalink] 18 Mar 2017, 08:03
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