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Take x = 2

A. \(\frac{1}{x}\) = \(\frac{1}{2}\)

B. \(\frac{1}{1+x}\) = \(\frac{1}{3}\)

C. \(\frac{x}{1+x}\)= \(\frac{2}{3}\)

D. \(\frac{x}{\frac{1}{1+x}}\) = 6, this is greatest

E. \(\frac{x}{\frac{x}{1+x}}\) = 3
Bunuel
­If x is a positive integer greater than 1, which of the following has the greatest value?

A. \(\frac{1}{x}\)

B. \(\frac{1}{1+x}\)

C. \(\frac{x}{1+x}\)

D. \(\frac{x}{\frac{1}{1+x}}\)

E. \(\frac{x}{\frac{x}{1+x}}\)


­
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Bunuel
­If x is a positive integer greater than 1, which of the following has the greatest value?

A. \(\frac{1}{x}\)

B. \(\frac{1}{1+x}\)

C. \(\frac{x}{1+x}\)

D. \(\frac{x}{\frac{1}{1+x}}\)

E. \(\frac{x}{\frac{x}{1+x}}\)
Let's try without substituting values
We know: x > 1
A. 0 < 1/x < 1
B. 1/(1+x) - this has a larger denominator compared to 1/x, hence is even smaller
C. x/(1+x) = 1/(1/x + x/x) = 1/(1/x + 1) - clearly, this is smaller than 1 since the denominator is greater than the numerator
D. x/[1/(x+1)] = x(x+1) > 1
E. x/[x/(x+1)] = x+1

Clearly, x(x+1) > x+1 since x>1

Ans D
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