Existential proposition plays an important role in mathematics. Knowing that something exists, even if you don't know what it is, is also a kind of information; There is sometimes a big difference between being and not being. According to the existence of that point, the Lagrange mean value theorem mentioned in the title can be used to derive L'H?pital's law-it should be the standard teaching method in many textbooks. The key point here is that the point where the median is located is between the intervals (a, b). Besides, you can do many things, and you don't know where it is and what the expression is.

In fact, many things in modern mathematics are "when you know they exist, it is difficult to write concrete expressions". On the one hand, it is difficult to write explicitly. To put it bluntly, mathematicians are not competent enough at present. On the other hand, just knowing existence is enough to do many things. For example, if we only know that there is a metric with non-negative curvature but not constant 0 on a closed surface, we can immediately deduce that the genus of the surface is 0 without knowing the specific expression (gauss-bonnet) of this metric; Many inequalities in PDE are "there is a constant c, which makes the following inequality hold", but people usually don't care too much about the specific expression of this constant (but what quantity this constant depends on still needs to be written). Qiu Chengtong proved Calabi's conjecture with the important work of Fields Prize, which actually proved the existence of solutions of partial differential equations. In addition, there is not much information-there may be some estimates about the size and growth of the solution, but the specific expression of the solution is absolutely not available, and it is probably impossible to write it. But this alone is enough for him to win the most important prize in the field of mathematics, and it also provides a suitable string theory model for theoretical physicists.

But in many cases, existence is only the first step in mathematical research, which does not mean that mathematicians will ignore everything after proving existence. Mr. Qiu's work mentioned above actually created a field called Calabi-Yau geometry, in which many mathematicians are doing something more specific than existence-for example, studying the modular space composed of all Calabi-Yau metrics. Nothing can be done overnight. Many times, you have to take it step by step. The existence of the classical solution of Naville-Stokes equation in the seven difficult problems of the Millennium is still unknown.