One of the example in
Rolle's Theorem is
a car driving at an uniform speed, its acceleration is 0. It can also
be stated that if a car has zero acceleration, then its velocity is
uniform. This is true in the physics, and it is
an example of how the theorem works. The theorem states:
If the derivative
of a function at all points are 0 in an open interval (a, b), then
the function is constant on (a, b).
Is there any case that this theorem fails? Consider another
example,
The function is obviously not a constant. Does theorem
fail in this function? The answer is no. Since when x
equals 0, this function is not differentiable. In (0, +∞) and (
∞,
0)
the function are constant.
