A Bound on the Second-Order Taylor Remainder by the Second and Third Derivatives

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f$ and its successive derivatives $f'$ and $f''$ are differentiable, and let $M_{2},M_{3}\in\mathbb{R}$ satisfy $|f''(x)|\le M_{2}$ and $|f'''(x)|\le M_{3}$ for every $x\in\mathbb{R}$.

(i) Second-order remainder. For all $v,h\in\mathbb{R}$,

$$\Bigl|f(v+h)-f(v)-f'(v)h-\tfrac12f''(v)h^{2}\Bigr|\le\min\bigl\{M_{2}h^{2},\tfrac12M_{3}|h|^{3}\bigr\}.$$