The linearization $L(x) = f(a) + f'(a)(x-a)$ is exact for every function on a small enough interval.

False

False. It is exact only where the error term $\tfrac12 f''(c)(x-a)^2$ vanishes, that is, for linear functions. For everything else it is an approximation whose error grows with the square of the distance, at a rate set by the curvature. Small intervals make it good, never exact.