Question

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

Answer 1

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