noboru wrote:

If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?

I. \(\frac{\sqrt{x+y}}{2}\)

II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)

III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)

A. I only

B. II only

C. III only

D. I and II only

E. None

Let's

test some values.x = 1 and y = 11/√(x + y) = 1/√(1 + 1) =

1/√2I. √(x + y)/2 = √(1 + 1)/2 = √2/2

Notice that, if we take

1/√2 and multiply top and bottom by √2, we get: √2/2, which is the same as quantity I

Since quantity I is not greater than

1/√2,

statement I is not trueII. (√x + √y)/2 = (√1 + √1)/2 = (1 + 1)/2 = 2/2 = 1

Since 1 IS greater than

1/√2, we cannot say for certain whether quantity II will

always be greater than √(x + y)/2

III. (√x - √y)/(x + y) = (√1 - √1)/(1 + 1) = (1 - 1)/2 = 0/2 = 0

Since 0 is not greater than

1/√2,

statement III is not trueSo, statements I and III are definitely not true, and we aren't yet 100% certain about statement II

Let's try another pair of values for x and y

x = 0.25 and y = 0.251/√(x + y) = 1/√(0.25 + 0.25) =

1/√0.5Let's further simplify

1/√0.5Since 1 = √1, we can say:

√1/√0.5Then we'll use a rule that says (√k)/(√j) = √(k/j)

So,

√1/√0.5 = √(1/0.5) = √2We see that, when x = 0.25 and y = 0.25, 1/√(x + y) =

√2II. (√x + √y)/2 = (√0.25 + √0.25)/2 = (0.5 + 0.5)/2 = 1/2

Since 1/2 is NOT greater than

√2,

statement II is not trueAnswer:

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