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$ {\text{Solution: - }} \\

{\text{To check a function either it is increasing or decreasing we have to double differentiate the function}} \\

{\text{and check the function in their domain either it is increasing or decreasing}} \\

{\text{in this question our function is }}f(x) = {x^3} - 1 \\

{\text{ so let's find the first derivative }}f'(x) = 3{x^2} \\

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

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

{\text{II}}{\text{. To find the root of }}f'(x){\text{ we have to equate }}f'(x) = 0. \\

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

{\text{there is one root of }}f'(x){\text{ in ( - 1,1]}}{\text{.}} \\

\therefore {\text{Statement I is correct and II is incorrect }} \\

{\text{Answer is A}}{\text{.}} \\

{\text{Note: - To check a function either it is increasing or decreasing we have to differentiate the function}} \\

{\text{ when first derivative is always positive in the given domain then it is strictly increasing}}{\text{.}} \\

{\text{ }} \\

$

{\text{To check a function either it is increasing or decreasing we have to double differentiate the function}} \\

{\text{and check the function in their domain either it is increasing or decreasing}} \\

{\text{in this question our function is }}f(x) = {x^3} - 1 \\

{\text{ so let's find the first derivative }}f'(x) = 3{x^2} \\

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

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

{\text{II}}{\text{. To find the root of }}f'(x){\text{ we have to equate }}f'(x) = 0. \\

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

{\text{there is one root of }}f'(x){\text{ in ( - 1,1]}}{\text{.}} \\

\therefore {\text{Statement I is correct and II is incorrect }} \\

{\text{Answer is A}}{\text{.}} \\

{\text{Note: - To check a function either it is increasing or decreasing we have to differentiate the function}} \\

{\text{ when first derivative is always positive in the given domain then it is strictly increasing}}{\text{.}} \\

{\text{ }} \\

$

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