\[ {\text{Consider the following statements in respect of the function }}f(x) = {x^3} - 1, \\
  x \in [ - 1,1] \\
  {\text{I}}{\text{. }}f(x){\text{ is increasing in }}[ - 1,1] \\
  {\text{II}}f'(x){\text{ has no root in (}} - 1,1]. \\
  {\text{Which of the statements given above is/are correct?}} \\
  {\text{A}}{\text{. only I}} \\
  {\text{B}}{\text{. only II}} \\
  {\text{C}}{\text{. Both I and II}} \\
  {\text{D}}{\text{. Neither I nor II}} \\
\]

$ {\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{ }} \\
$