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

2

Explanation

We have f(X)=`(x+1)^(1//3) -(x-1)^(1//3)`
`therefore f(X)=1/3(x+1)^(-2)/(3)-(1)/(3)(x-1)^(-1)/(3)`
`=((x-1)^(2//3)-(x+1)^(2//3))/(3(x^(2)-1)^(2//3))`
Clearly f(X) does not exist at x`=pm 1`now
f(x)=0
or `(x-1)^(2//3)=(x+1)^(2//3)`
or `(x-1)^(2)(x+1)^(2)`
or -2x=2xor 4x =0 or x=0
Clearly `f(x) ne` 0 for any other values of x in [0,1]
The value fo f(x) at x=0 is 2 ltrbgt Hence the greatest value of f(x) is 2