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

Question

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

Vedantu Content Team · Accepted Answer

$  {	ext{Solution: - }}   \  {	ext{To check a function either it is increasing or decreasing we have to double differentiate the function}}   \  {	ext{and check the function in their domain either it is increasing or decreasing}}   \  {	ext{in this question our function is }}f(x) = {x^3} - 1   \  {	ext{ so let's find the first derivative }}f'(x) = 3{x^2}   \  {	ext{Now the second derivative is }}f''(x) = 6x{	ext{ , we check the function for }}x \in [ - 1,1]   \  {	ext{here }}f''(x) \in [ - 6,6]{	ext{   }}	herefore {	ext{ the function }}f(x){	ext{ is increasing }}{	ext{.}}   \  {	ext{II}}{	ext{. To find the root of }}f'(x){	ext{ we have to equate }}f'(x) = 0.   \   \Rightarrow 3{x^2} = 0{	ext{   }} \Rightarrow x = 0{	ext{ }}   \  {	ext{there is one root of }}f'(x){	ext{ in ( - 1,1]}}{	ext{.}}   \  	herefore {	ext{Statement I is correct and II is incorrect }}   \  {	ext{Answer is A}}{	ext{.}}   \  {	ext{Note: -  To check a function either it is increasing or decreasing we have to differentiate the function}}   \  {	ext{           when first derivative is always positive in the given domain then it is strictly increasing}}{	ext{.}}   \  {	ext{           }}   \ $

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