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Weekly Challenge 30: the Sign of the Times

Stage: 5 Short Challenge Level: Challenge Level:2 Challenge Level:2

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?

Did you know ... ?

Problems in calculus can often be considered from either an algebraic or geometric viewpoint and calculus is fundamental in the advanced study of geometry as well as areas of theoretical physics, such as string theory and relativity.