In Rolls theorem f(x) is continuous on [a,b] but f(x) is differentiable on (a,b) my question is that why f(x) is not differentiable on [a,b] that is on closed interval.

 

Asked by Kalyanrao Chavan | 2nd Aug, 2014, 10:11: PM

Expert Answer:

L e t space f open parentheses x close parentheses space b e space a space r e a l space v a l u e d space f u n c t i o n space d e i n e d space o n space open parentheses a comma b close parentheses space a n d space l e t space c element of open parentheses a comma b close parentheses comma space t h e n space a space f u n c t i o n space i s space d i f f e r e n t i a b l e space i f space a n d space o n l y space i f space limit as x rightwards arrow c of fraction numerator f open parentheses x close parentheses minus f open parentheses c close parentheses over denominator x minus c end fraction space e x i s t s space f i n i t e l y. S u p p o s e space o n space t h e space c o n t r a r y comma space i f space t h e space f u n c t i o n space i s space d e f i n e d space o n space t h e space c l o s e d space i n t e r v a l open square brackets a comma b close square brackets. B y space t h e space d e f i n i t i o n space o f space d i f f e r e n t i a b i l i t y comma space w e space h a v e comma space f open parentheses x close parentheses space i s space u n d e f i n e d space a n d space h e n c e limit as x rightwards arrow a to the power of minus of fraction numerator f open parentheses x close parentheses minus f open parentheses a close parentheses over denominator x minus a end fraction space i s space u n d e f i n e d space i f space x less than a T h e r e f o r e space t h e space d e r i v a t e space c a n n o t space e x i s t space a t space a. T h i s space i s space t h e space m a i n space r e a s o n space t h a t space w e space a r e space c o n s i d e r i n g space t h e space f u n c t i o n space i s space d i f f e r e n t i a b l e space i n space t h e space o p e n space i n t e r v a l.

Answered by Vimala Ramamurthy | 4th Aug, 2014, 10:26: AM