Proving a Function is Less than a Value

Question

I'm working on a multipart problem and I've been asked to prove that a value W < 0.5.

I worked out W and reached $$W = \frac{\sqrt{X1*X2*Y1*Y2}}{X1*Y1+X2*Y2+X1*Y2} $$

but I'm not sure how to progress from here to prove that W < 0.5. Though I placed values into the equation using excel and can see that it will indeed be a fraction.

Answer 1

Divide the top into each term in the bottom and let a = sqrt(X1 Y1 / X2 Y2). Let b = sqrt( X1 Y2 / X2 Y1). Then W becomes 1/(a + 1/a + b) which is less than 1/(a + 1/a) which is at most 1/2 since the minimum of the function x + 1/x is 2 for x > 0.