All possible value of `f(x)=(x+1)^(1/3)-(x-1)^(1/3)` on [0,1] is 1 (b) 2 (c) 3 (d) `1/3`

Options

A. 1
B. 2
C. 3
D. `1//3`

Answer

B

Explanation

We have
`f(x)=(x+1)^(1//3)-(x-1)^(1//3)`
`rArrf'(x)=1/3{1/(x+1)^(2//3)-1/(x-1)^(2//3)}=((x-1)^(2//3)-(x+1)^(2//3))/(3(x^2-1)^(2//3))`
Clearly ,f'(x) does not exist at`x+-1`
Now `f'(x)=0rArr(x-1)^(2//3)=(x+1)^(2//3)rArrx=0`
Clearly f'(x)`ne` 0 for any other value of `x in [0,1]`
The value of f(x)at x=0 and=d x=1 are 2 and `2^(1//3)` respectively
Hence the greatest value of f(x) is 2