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In [0, 1] Lagrange's mean value theorem ...

In `[0, 1]` Lagrange's mean value theorem is not applicable to

A

`f(x)={((1)/(2)-x",", x lt (1)/(2)),(((1)/(2)-x)^(2)",",x ge (1)/(2)):}`

B

`f(x)={((sinx)/(x)",",x ne 0),(1",",x=0):}`

C

`f(x)=x|x|`

D

`f(x)=|x|`

Text Solution

Verified by Experts

The correct Answer is:
A
There is only one function in option (A), whose critical point `(1)/(2) in (0,1)` but in other parts critical point `0 notin(0,1)`. Then we can say that functions in options (b), (c) and (d) are continuous on [0, 1] and differentiable in (0, 1).
Now, for `f(x)={((1)/(2)-x",",x lt (1)/(2)),(((1)/(2)-x)^(2)",", x ge (1)/(2)):}`
Here `Lf'((1)/(2))=-1 and Rf' ((1)/(2))=2 ((1)/(2)-(1)/(2))(-1)=0 therefore Lf'((1)/(2) ne Rf'((1)/(2))`
`rArr f` is non differentiable at `x=(1)/(2) in (0, 1) therefore `LMVT is NOT applicable to f(x) in [0, 1]
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