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 In the mean value theorems, we discuss the continuity of a given function in the closed interval and the differentiability in the open interval. Why we don’t discuss its differentiability in the closed interval?

yeh is liay k aik function [a,b] k closed interval par continuous tou ho sakta hai, par [a,b] par differentiable nahin ho sakta...
example hai y = sqrt(1-x^2)
yeh function x = -1 and 1 par continuous hai....par -1 aur 1 par differentiable nahin hai....

yaad rakhain k mean value theorem mein hamain function k [a,b] k closed interval par differentiate honay ya nah honay se koi gharz nahin hoti....sirf [a,b] par continuous hona hi kafi hota hai...kyun k mean value theorem k mutabiq agar yeh teen cheezain satisfy hon
1. function is continuous on the closed interval [a,b].
2. function is differentiable on the open interval (a,b).
tou [a,b] interval k darmiaan aik aisa point c hota hai jahan par instantaneous rate of change at that point, equal hota hai, overall average rate of change k, in the interval (a,b).
average rate of change ka formula hai [ f(b)-f(a) ] / (b-a)...
jab k point c choon k [a,b] interval k darmiaan lie karta hai aur function hamara continous hai tou point c par function differentiable bhi ho ga...
is se yehi pata chalta hai k [a,b] par function k differentiable honay se hamain koi gharaz nahin hai....agar function end points par differentiable nah bhi ho tou tab bhi hamain koi faraq nahin paray ga aur ham mean value theorem apply kar sakain gay....

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