In particular the proof is incorrect Find a counterexample t

In particular, the proof is incorrect. Find a counterexample that demonstrates the falsity of the statement. Then identify the flaw(s) in the argument. Is the following a proof? If not, why not? Read it through and carefully explain to yourself how each line follows (or fails to follow) from the assumptions and the previous lines in the proof. Theorem: For all real numbers x, y, 1/2 (x + y) greaterthanorequalto squareroot xy. Proof: Squaring and multiplying through by 4, x^2 + 2xy + y^2 squareroot 4xy, so subtracting 4xy from each side, x^2 - 2xy + y^2 squareroot 0. But x^2 - 2xy + y^2 = (x - y)^2 which is always squareroot 0, so the theorem is proved.

Solution

sqrt{xy} is defined only for xy>=0

If xy<0 then we would have sqrt{xy}=isqrt{|xy|}