At local minima/maxima, aren't functions constant for an infinitesimally small amount of time?

Question

If you are looking at a graph of sine describing an object's motion at $\frac{\pi}{2}$, where the derivative is 0, doesn't this object have a constant position for an instant? I was thinking about how horizontal lines have zero slopes, which represent constant position in the physics example, and therefore have a derivative of zero. But these local maxima and minima have a derivative of zero at an instant, so it leads me to infer that they are constant along an infinitesimally small interval, or at a point. I wanted to describe the point in which it's constant as being multiples of $\frac{\pi}{2}$. But my teacher said that only if the derivative is 0 at more than 1 point. Or am I approaching this the wrong way?