Consider the following statements in respect of the function $f(x) = x^{3} - 1, \quad x\epsilon [-1, 1]$

I. $f(x)$ is increasing in $[-1, 1]$

II. $f'(x)$ has no root in $(-1, 1)$.

Which of the statements given above is/ are correct?

Given the function $f(x) = x^{3} - 1, \quad x\epsilon [-1, 1]$

$f'(x)=3x^2$ which is positive for all $x$

and at $x=0$,

$f'(x)=0$,

$f''(x)=0$

I. $f(x)$ is increasing in $[-1, 1]$

$f(-1)=-2$ and

$f(1)=0$

Thus

$f(x)$ is increasing in the given interval.

II. $f'(x)$ has a root , $x=0$ in $(-1, 1)$.

$\therefore$only I is correct.