If X and yard positive numbers, show that (x+y)(1/x + 1/y)>4

Answer 1

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.