\[ {\text{Consider the following statements in respect of the function }}f(x) = {x^3} - 1, \\
x \in [ - 1,1] \\
{\text{I}}{\text{. }}f(x){\text{ is increasing in }}[ - 1,1] \\
{\text{II}}f'(x){\text{ has no root in (}} - 1,1]. \\
{\text{Which of the statements given above is/are correct?}} \\
{\text{A}}{\text{. only I}} \\
{\text{B}}{\text{. only II}} \\
{\text{C}}{\text{. Both I and II}} \\
{\text{D}}{\text{. Neither I nor II}} \\
\]
Answer
370.2k+ views
$ {\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{ }} \\
$
Last updated date: 02nd Oct 2023
•
Total views: 370.2k
•
Views today: 3.70k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE
