Solution
= Solution
By <Taylor's theorem> with <Lagrange remainder>, for an increment $t$,
$$
\boxed{f(x+t)=f(x)+tf'(x)+\frac{t^2}{2}f''(x)+R_x(t),\qquad R_x(t)=\frac{t^3}{6}f'''(\xi_t),}
$$
where $\xi_t$ lies between $x$ and $x+t$. The given third <derivative> bound implies $|R_x(t)|\le |t|^3/6$.