# ${\text{Consider the following statements in respect of the function }}f(x) = {x^3} - 1, \\ x \in [ - 1,1] \\ {\text{I}}{\text{. }}f(x){\text{ is increasing in }}[ - 1,1] \\ {\text{II}}f'(x){\text{ has no root in (}} - 1,1]. \\ {\text{Which of the statements given above is/are correct?}} \\ {\text{A}}{\text{. only I}} \\ {\text{B}}{\text{. only II}} \\ {\text{C}}{\text{. Both I and II}} \\ {\text{D}}{\text{. Neither I nor II}} \\$
${\text{Solution: - }} \\ {\text{To check a function either it is increasing or decreasing we have to double differentiate the function}} \\ {\text{and check the function in their domain either it is increasing or decreasing}} \\ {\text{in this question our function is }}f(x) = {x^3} - 1 \\ {\text{ so let's find the first derivative }}f'(x) = 3{x^2} \\ {\text{Now the second derivative is }}f''(x) = 6x{\text{ , we check the function for }}x \in [ - 1,1] \\ {\text{here }}f''(x) \in [ - 6,6]{\text{ }}\therefore {\text{ the function }}f(x){\text{ is increasing }}{\text{.}} \\ {\text{II}}{\text{. To find the root of }}f'(x){\text{ we have to equate }}f'(x) = 0. \\ \Rightarrow 3{x^2} = 0{\text{ }} \Rightarrow x = 0{\text{ }} \\ {\text{there is one root of }}f'(x){\text{ in ( - 1,1]}}{\text{.}} \\ \therefore {\text{Statement I is correct and II is incorrect }} \\ {\text{Answer is A}}{\text{.}} \\ {\text{Note: - To check a function either it is increasing or decreasing we have to differentiate the function}} \\ {\text{ when first derivative is always positive in the given domain then it is strictly increasing}}{\text{.}} \\ {\text{ }} \\$