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'Turning to Calculus' printed from http://nrich.maths.org/
The gradients and signs of functions are strongly related for any
nice, smooth curve which you can draw. To get into the problem you
can sketch small line segments and ask yourself: how can these be
smoothly joined together?
Complications can arise for curves which turn suddenly or have gaps
or become infinite at various points. Maybe you could think about
creative ways to get from one place to the next.