This problem is good for 'simmering' in the background. If you can't make sense of it at first, try leaving it and coming back to it later.

My friend has sketched a non-constant curve $f(x)$ which passes through the origin. She knows that its derivative exists at all points. Is it possible that $f(x)$ could satisfy
$$
f(x)\times\frac{df(x)}{dx} \leq 0
$$ for all $x$?
 
I'll need a very clear explanation to convince me!
 
My other friend has sketched a curve whose derivative exists at all points, but his does not pass through the origin. Is it possible that the same condition might hold?