I need a prove of this result , it a NCERT question .if f(x) is continous in [a,b], and f'(x)>0 in (a,b), Prove that f(x) is increasing in [a,b].( please note the closed nature of the interval ) i feel this is a application of langrange mean value theorem or intermediate theorem. PLease help

As f(x) is both continuous and differential...we can apply lagranges theorem

by formula it is sure that for f'(x) to be >0 , b-a should be >0...............

So,a is at lower pt. and b is at higher point..............and hence a increasing function......

f'(x)=f'(b)-f'(a)/b-a....

You are right b-a>0

but how u reach this point that b is a higher point ( i mean f(b) ) and a is a lower point ,

You are inherently using the result that if f'(x)>0 the function is increasing

hence , b is a higher point and a is a lower point

and how come

f'(x)= (f'(b)-f'(a) )/ (b-a) ???