Consider the function y = x2 with it's domain restricted [0, 2), we know that the function is monotonic for this interval.
Now, Mean value theorem's corollary 3 gives us that if f'(x) > 0, then the function is increasing on the interval (a, b). It's on (a, b) because, the derivative is not defined on the endpoints.
Now, using this corollary, If f''(x) >0, then f'(x) also must be increasing. This must be the case that f''(x) must be defined for the open interval of (a, b). Letting dx define the open interval as: (a + dx, b - dz), then f'(x) must be increasing for this interval, which would define the concavity of the curve.
But, the problem is, what would the function be defined as, either increasing or decreasing, within difference of these intervals i.e. on (dx, dz). I've pictorially illustrated this below: (even though I've visualized this for y = x2 I'm referring to curves in general)
