We are given that x and y are positive real numbers. We need to show that the following inequality holds:
(x+y)(1/x + 1/y) > 4
Expanding the left side, we get:
x/x + x/y + y/x + y/y
Simplifying this, we get:
2 + (x/y + y/x)
Now, by the AM-GM inequality, we know that:
(x/y + y/x) ≥ 2
Therefore:
2 + (x/y + y/x) > 4
And we have shown that:
(x+y)(1/x + 1/y) > 4
Thus, the proof is complete.
If X and yard positive numbers, show that (x+y)(1/x + 1/y)>4
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