Is truncated taylor series a bound?

Question

Given a function $f(x)$ with truncated Taylor series $T_n(x)$ at point $x = 0$, it's possible to $T_n(x)$ cross $f(x)$ in a point $x \neq 0$? In other words, is the truncated Taylor series always an upper/lower for the original function for each interval $x > 0$ and $x > 0$? There's a known result about it? I know that at least the functions can coincide (for instance, the first Taylor series term of $\cos{(x)}$, at multiples of $2\pi$).