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Statement-1 : If f is differentiable on ...

Statement-1 : If f is differentiable on an open interval (a,b) such that `|f'(x)le M "for all" x in (a,b)`, then `|f(x)-f(y)|leM|x-y|"for all" in (a,b)`
Satement-2: If f(x) is a continuous function defined on [a,b] such that it is differentiable on (a,b) then exists `c in` (a,b) such that
`f'(c)=(f(b)-f(a))/(b-a)`

A

Statement-1 is True, Statement-2 is Ture, Statement-2 is a correct explanation for statement-1

B

Statement-1 is True, Statement-2 is Ture, Statement-2 is not a correct explanation for statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 True.

Text Solution

Verified by Experts

The correct Answer is:
A
Clearly, statement-2 is true. (Statement of Largrange's Mean Value Theorem)
Let `x , y in (a,b) "such that" xlty`
Applying Largrange's mean value theorem on [x,y] we get `'f(c)=(f(x)-f(x))/(x-y)"for some" c in [x,y]`
`rArr |f'(c)|=(|f(x)-f(y)|)/(|x-y|)`
`rArr(|f(x)-f(y)|)/(|x-y|)leM`
`rArr|f(x)-f(y)|leM|x-fy|`
So, statement-1 is true. Also, statement-2 is correct explantion for statement-1
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