tag:blogger.com,1999:blog-9004452969563620551.post5127502962885103231..comments2018-09-18T12:34:27.340+05:30Comments on Paramanand's Math Notes: Monotone Functions: Part 2testernoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-9004452969563620551.post-21358453698916106022013-03-20T11:38:51.950+05:302013-03-20T11:38:51.950+05:30Hello Arnab,
I read your article and what you have...Hello Arnab,<br />I read your article and what you have mentioned is correct. If we present only the statement of Rolle’s and LMVT to any reasonable person he will at once say that Rolle’s is a special case of LMVT when the function values at the end points of the interval are same. However if we wish to establish these theorems there is no way to prove LMVT without using the Rolle’s.<br /><br />Most importanty. from the graphical point of view these theorems have the same content. If there is a graph for a function satisfying Rolle’s condition, you just need to rotate the X-Y axes and the graph now becomes applicable to the LMVT. In Rolle’s the secant is parallel to X-axis, but in LMVT it is not necessarily parallel to X-axis.<br /><br />As far as I am aware I have not seen any proof of LMVT without using Rolle’s Theorem. If you have any references regarding this you can let me know.<br /><br />And by the way I have not touched upon Cauchy’s MVT but just to add this is a further generalization and it can also be proved using Rolle’s Theorem. And more advanced results like Taylor’s / Maclaurin’s theorem are also proved using Rolle’s theorem. By choosing functions suitably to meet the hypothesis of Rolle’s theorem, many advanced theorems can be proved. So in the theory of mean value theorems Rolle’s is the most fundamental one.<br />paramanandhttps://www.blogger.com/profile/03855838138519730072noreply@blogger.comtag:blogger.com,1999:blog-9004452969563620551.post-46750209373765292292013-03-20T11:38:10.749+05:302013-03-20T11:38:10.749+05:30Great work..what an article..!!!
But I would like...Great work..what an article..!!!<br /><br />But I would like to seek Your attention towards one issue..<br /><br />You’ve written that Mean Value Theorem can be derived from Rolle’s Theorem itself..but I think what I’ve learned is completely the opposite to that..which I’ve also mentioned in an article written by myself..here..<br /><br />http://cosmologistic.blogspot.in/2012/10/lagranges-mvt-for-beginners.html<br /><br />But Yes LMVT can for sure be derived from CAUCHY’s Theorem..is that what You wanted to tell ??Arnab(@carmenelo)http://twitter.com/carmenelonoreply@blogger.com