in a smooth curve (no pointed or sharp or edge-y bending) of s-t (displacement-time graph)... will there , at least, be one point where instantaneous velocity = average velocity (up to that time)?
what if the curve is not smooth? ie it has pointed shape like spikes..?

Question

in a smooth curve (no pointed or sharp  or edge-y bending) of s-t (displacement-time graph)... will there , at least, be one point where instantaneous velocity = average velocity (up to that time)?
what if the curve is not smooth? ie it has pointed shape like spikes..?

anonymous · Answer

Yes.  The mean value theorem guarantees that a smooth, continuous displacement-time graph will have at least one point where the instantaneous velocity equals the average velocity for the trip.  If the graph has corners or kinks in it, it will not be differentiable at these points so we cannot apply the mean value theorem at these points.

yash2651995 · Answer

ithanks.. i got ya!!
i wasnt clear about kink's part but now i'm clear why there necessarily wont be any such Vinstant= v avg case 
THANKS AGAIN :)