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If nature selects mean character value, ...

If nature selects mean character value, it is

A

`f(x)={{:(1/2-x",",xlt 1/2),((1/2 -x)^2",", xge1/2):}`

B

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

C

`f(x) = x absx`

D

`f(x) = absx`

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 criticle point `0cancelin(0,1)`. Then we can say that functions in options (b), (c ) and (d) are continuous on [0,1] and defferentiable in (0.1)
Now , for `f(x)={{:(1/2-x",",x lt 1/2),((1/2-x)^2",",x ge1/2):}`
Here `Lf.(1/2) = -1 and Rf. (1/2) =2(1/2-1/2) (-1) =0 :. Lt. (1/2 ) ne Rf. (1/2)`
`rArr` f is non differentiable at `x = 1/2 in (0,1) :. ` LMVT is NOT applicable to f(x) in [0,1]
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