Question

# Consider the following statements in respect of the function $f(x) = {x^3} - 1$$x \in [ - 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?A. only IB. only IIC. Both I and IID. Neither I nor II

Hint: Differentiate the given function twice and find whether f(x) is increasing by substituting [ - 1,1] and differentiate the function once and substitute ( - 1,1] to find whether the function has roots.

I. To check a function either it is increasing or decreasing we have to double differentiate the function and check the function in their domain either it is increasing or decreasing in this question our function is $f(x) = {x^3} - 1$

So let's find the first derivative $f'(x) = 3{x^2}$

Now the second derivative is $f''(x) = 6x$, we check the function for $x \in [ - 1,1]$

here $f''(x) \in [ - 6,6]$ $\therefore$ the function f(x) is increasing.

II. To find the root of $f'(x)$ we have to equate $f'(x) = 0$.

$\Rightarrow 3{x^2} = 0{\text{ }} \Rightarrow x = 0{\text{ }}$

there is one root of $f'(x)$ in ( - 1,1].

$\therefore$ Statement I is correct and II is incorrect