Origin of the upper limit: Fan Zeng's Song Tomb in Daping Forest Farm, Shunchang, Fujian: "The Japanese Kamakura era is about equivalent to the Southern Song Dynasty in China, so the upper limit of Tongan kiln era was pushed to the Southern Song Dynasty."
Lower bound: generally refers to the minimum value of a function or the minimum value of an independent variable. In mathematical analysis, in a given range (relative extreme value) or the whole definition domain of a function (global or absolute extreme value), the maximum and minimum values of a function are collectively called extreme values (poles).
Lower bound application: find the maximum and minimum value of a function
Finding the global maximum and minimum is the goal of mathematical optimization. If the function is continuous in the closed interval, there are global maxima and minima through the maximum theorem. In addition, the global maximum (or minimum) must be a local maximum (or minimum) within the domain, or it must be located on the boundary of the domain. So the way to find the global maximum (or minimum) is to look at all the local maximum (or minimum) inside, and also look at the maximum (or minimum) of the points on the boundary, and take the maximum or minimum.
Fermat's theorem can find the differential function of local extreme value, which shows that it must happen at the critical point. We can use the first derivative test, the second derivative test or the higher derivative test to distinguish whether the critical point is a local maximum or a local minimum, and give sufficient distinguishability.
For any function defined by a segment, find the maximum (or minimum) by finding the maximum (or minimum) of each part separately, and then see which is the maximum (or minimum).