Once upon a time in 2017 a friend asked me to explain the mumbo-jumbo of a mathematical article called "New Multidirectional Mean Value Inequality". Since this happened on a social media, I can show the hilarious record. I tried to mimic Richard Feynman to see if I can do like him. The result left me with mixed feelings as to the value of academic work. Explaining it was a good fun though!
'Do you really want to call forth one hell of a long-winded nutty professor??! LOL'
Okay... What would you say about the following statement: One drives from point A to point B and we monitor its speedometer. At the end his or her AVERAGE SPEED is x km/h, because he or she drove a hours and the distance between A and B is ax kilometers. Is it obvious that at some point in time (say, at some second) his or her speedometer must have been pointing at EXACTLY x km/h?
Whatever answer. I am just accessing the common ground. There is nothing tricky in the question, even though my inner reptile may take some delight in me parading as school teacher. ;)
So, this is Lagrange Theorem considered as one of the cornerstones of Differential Calculus! My students will be proving it big time on Friday, because they chose to miss the lecture and I will be having my sweet revenge. LOL
It is worth noting that Lagrange Theorem has "skinny" French flavour. In particular it does not transfer to discrete setting. For example, I my have gained 10$ Monday (say, winning a bet on some of the weekends football games) and then spent each day some change on coffee and tea, and be even by Sunday: meaning my average rate of spending is zero, but this does not mean there have been a day without some spending, even if it was negative spending, as on, say, Tuesday.
Most people wonder how Lagrange Theorem is applicable at all, since we have no information on when exactly that moment(s) will be when the speedometer will also show the average speed. So, mostly Lagrange Theorem is used in situations where a priory information about all instant speeds is available. For example, if the driver drives ALL THE TIME at speed greater than x km/h then he or she will make the distance in less than a hours. This statement can be called 'differential inequality', because in this interpretation differential is simply speed and we have > and < instead of =. Note that this statement (the differential inequality) is much more ROBUST in that it can be transferred to many settings, like discrete times or even psychological: if one is constantly of good motives then his or her deeds will be good.
We are almost there. :)
Having explained the words "differential" and "inequality" in the title, there a couple more. Now "new" means simply that the research is original, that is, no-one have proved this particular form before. About "multidirectional"...
(About multidirectional we'd rather skip here.)
That's all. LOL!
Great many thanks for asking! I did not know this explanation few days ago.
As some nutty professors like to joke: "I explained it so well, that even I grasped it!" LOL!
*Image by Ralf Kunze from Pixabay