Algebra..#1

Algebra Level 1

Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y.

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The answer is 2.

Harshvardhan Mehta by Harshvardhan Mehta , 23, India

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3 solutions

Aditya Raman
Jun 6, 2014

AM > GM => (x+y)/2 > root over (xy) => x+y > 2 × root over (1) Therefore minimum value of x+y is 2.

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Can someone tell me what's the significance of x> 0 ?

Aditya Raman - 7 years ago

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It tells us that x is positive

Vighnesh Raut - 7 years ago

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Not just that. AM-GM only applies to non-negative reals, so this condition allows you to apply AM-GM correctly.

If the question was not restricted to x > 0 x > 0 , then there is no minimum value. E.g take x = − 1 0 n , y = − 1 0 − n x = - 10^n, y = - 10^{-n} .

Calvin Lin Staff - 7 years ago

Because x=y= -1 would produce xy=1 and x+y = -2

Greg Grapsas - 2 years, 3 months ago
Vighnesh Raut
Jun 6, 2014

xy=1

∴ x = 1 y \therefore \quad x=\frac { 1 }{ y }

Now, x+y is minimum only and only if x and y are integers.

This is only possible when y = 1. So, x= 1 & x+y(minimum) = 2

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Edwin Gray
Feb 26, 2019

If xy = 1, then x + y = x + 1/x. The derivative is 1 - (1/(x^2). If this = 0, then x = 1, and 1/x = 1, so x + y = 2. To show this is a minimum, note that the second derivative > 0.

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