The function `f(x)=x^(1/3)(x-1)` has two inflection points has one point of extremum is non-differentiable has range `[-3x2^(-8/3),oo)`

Options

A. has two inflection points
B. has one point of extremum
C. is non differentiable
D. has range `[-3xx2^(-8//3),oo)`

Answer

1,2,3,4

Explanation

`f(X)=x^(1//3)(x-1)`
or `d f(x)/(dx)=(4)/(3x^(1//3))-(1)/(3).(1)/(x^(2//3))=(1)/(3x^(2//3))[4x-1]`
Thus f(x) changes sign form -ve to +ve at `x=1//4 `which is point of minima
Also f(X) does not exist at x =0 as f(x) has veritcal tangent at x=0
`f(x)=4/9.(1)/(x^(2//3))+1/3.2/3.(1)/(x^(5//3))=(2)/(9x^(2//3))[2+(21)/(x)]`
`=(2)/(9x^(2//3))[(2x+1)/(x)]`
Thus f(x) =0 x=`1/2` which is the point of inflection at x = 0 f(x) does not exist but f(X) changes sign hence x =0 is also the point of inflection
From the above information the graph of y =f(x) is as shown

Also minimum value of f(x) is at x=`1//4` which is -`3xx2^(-8//3)` hence range is `[-3xx2^(-8//3),oo)`