For all real numbers x and y, $\displaystyle sqrt(xy) $ is less than or equal to
X + Y
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2
provide a counter example for statements that are false and provide a complete proof for those that are true
For all real numbers x and y, $\displaystyle sqrt(xy) $ is less than or equal to
X + Y
-----
2
provide a counter example for statements that are false and provide a complete proof for those that are true
Start with this: $\displaystyle (x-y)^2 \geq 0 \qquad \forall x, y \in \mathbb{R}^{\color{red}+}$
Playing around with it gives us:
$\displaystyle \begin{aligned}x^2 - 2xy + y^2 & \geq 0 \\ x^2 + 2xy + y^2 & \geq 4xy \\ (x+y)^2 & \geq 4xy \\ & \ \ \vdots\end{aligned}$
oh wow thanks
I was thinking so hard on that one too,
can you help me out with something else too i got a test in 2 hours xD
If we are asked to write the contrapostive of a conditional statement ( For each integer n, if $\displaystyle n^2$ is an odd integer, then n is an odd integer)
would the contra positive be replacing the odd's with even?