Inequalities

Algebra Level 3

If 0 < X < Y < 1 0< X <Y <1 , then which one of the following is not necessarily true?

(X^2 - Y)^2 < Y^2 X^2 < 2Y^2 (X - Y)^2 < X^2 X^3 - Y^3 < X^2

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Sudoku Subbu by sudoku subbu , 19, India

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3 solutions

Sarthak Singla
Sep 24, 2015

I have taken X = 0.1 and y = 0.2

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Michael Yi
Dec 29, 2014

Two of the easiest numbers to plug in would be X = 1/3 and Y = 2/3 Calculations become relatively simple, and when we get the statement (X-Y)^2 < X^2 We can see that it would turn out to be :

(1/3 - 2/3)^2 < (1/3)^2 (-1/3)^2 < (1/3)^2 1/9 < 1/9

Not true. Thus, the correct answer is: (X - Y)^2 < X^2

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Pranjal Jain
Dec 22, 2014

X = 0.1 , Y = 0.91 ( X − Y ) 2 = 0.81 and X 2 = 0.01 X=0.1, Y=0.91\\(X-Y)^{2}=0.81\text{ and } X^{2}=0.01

So, in this case ( X − Y ) 2 > X 2 (X-Y)^{2}>X^{2}

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