  Lower and upper bounds for a given function are important in many
mathematical and engineering contexts, where they often serve as a base for
both analysis and application. In this short paper, we derive piecewise linear
and quadratic bounds that are stated in terms of the Lipschitz constants of the
function and the Lipschitz constants of its partial derivatives, and serve to
bound the function's evolution over a compact set. While the results follow
from basic mathematical principles and are certainly not new, we present them
as they are, from our experience, very difficult to find explicitly either in
the literature or in most analysis textbooks.
