Question

If x^5y^4z^2 <0 , which of the following must be true?

I xy <0
II yz <0
III xz <0

Answer 1

$x^5y^4z^2 < 0\\\\y^4\geq0\ for\ y\in\mathbb{R}\\\\z^2\geq0\ for\ y\in\mathbb{R}\\\\x^5 > 0\ for\ y\in\mathbb{R^+}\ and\ x^5 < 0\ for\ x\in\mathbb{R^-}\\\\------------------------------\\\\x^5\underbrace{y^4z^2}_{> 0} < 0\iff x^5 < 0\Rightarrow xy < 0\ \wedge\ xz < 0\\\\------------------------------\\\\Answer:I\ xy < 0\ and\ III\ xz < 0.$