James Chen TMUA Paper 2 · Question 8

Consider the claim that $\frac{x+y}{2}>\sqrt{xy}$ for all positive $x,y$ . Here is an attempted proof of this claim:

\begin{aligned} (x-y)^2&>0 &&(1)\\ x^2+y^2-2xy&>0 &&(2)\\ x^2+y^2+2xy&>4xy &&(3)\\ \frac{(x+y)^2}{4}&>xy &&(4) \end{aligned}

Taking the square root of both sides, $\dfrac{x+y}{2}>\sqrt{xy}$ (5)

Which option best describes this attempt to prove the claim?

AThe claim is correct and the proof is correct

BThe claim is correct but the proof is incorrect

CThe claim & proof are incorrect and the first error is on line (1)

DThe claim & proof are incorrect and the first error is on line (2)

EThe claim & proof are incorrect and the first error is on line (3)

FThe claim & proof are incorrect and the first error is on line (4)

GThe claim & proof are incorrect and the first error is on line (5)

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