Answer
Verified
379.8k+ views
Hint: To check that the given function follows Rolle's Theorem or not, we will use the conditions of Rolle ’s Theorem one by one. If the given function satisfies all the conditions of Rolle’s Theorem, then the function holds the Rolle’s Theorem or not. And to check the converse of Rolle’s theorem, by using the condition of Rolle’s Theorem in reverse.
Complete step-by-step solution:
Since, the function is given in the question as:
$\Rightarrow f\left( x \right)=\left[ x \right]$ such that $x\in \left[ 5,9 \right]$
As we know that the greatest integer function is not a continuous function for any values belonging to $x$. We can clearly understand its diagram.
Since, the first condition of Rolle’s Theorem is that the function should be a continuous function over a closed interval, is not fulfilled by the given greatest integer function. So, we can say that the given greatest integer function does not satisfy the first condition of theorem, there is no need to proceed further to check another conditions of Rolle’s theorem to verify the function $f\left( x \right)=\left[ x \right]$ for$x\in \left[ 5,9 \right]$.
Since, the given function does not satisfy Rolle's Theorem. So, no need to check the converse of Rolle’s Theorem also.
Hence, the given function does not follow Rolle's Theorem.
Note: Here are the conditions of the Rolle’s Theorem as:
1. Function should be continuous on a closed interval.
2. Function should be differentiable on an open interval.
3. If $f\left( a \right)=f\left( b \right)$, there exists some $c$ belonging to the interval such that $f'\left( c \right)=0$.
If the function follows the first condition then go for second and third conditions.
Complete step-by-step solution:
Since, the function is given in the question as:
$\Rightarrow f\left( x \right)=\left[ x \right]$ such that $x\in \left[ 5,9 \right]$
As we know that the greatest integer function is not a continuous function for any values belonging to $x$. We can clearly understand its diagram.
Since, the first condition of Rolle’s Theorem is that the function should be a continuous function over a closed interval, is not fulfilled by the given greatest integer function. So, we can say that the given greatest integer function does not satisfy the first condition of theorem, there is no need to proceed further to check another conditions of Rolle’s theorem to verify the function $f\left( x \right)=\left[ x \right]$ for$x\in \left[ 5,9 \right]$.
Since, the given function does not satisfy Rolle's Theorem. So, no need to check the converse of Rolle’s Theorem also.
Hence, the given function does not follow Rolle's Theorem.
Note: Here are the conditions of the Rolle’s Theorem as:
1. Function should be continuous on a closed interval.
2. Function should be differentiable on an open interval.
3. If $f\left( a \right)=f\left( b \right)$, there exists some $c$ belonging to the interval such that $f'\left( c \right)=0$.
If the function follows the first condition then go for second and third conditions.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Who was the Governor general of India at the time of class 11 social science CBSE
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference Between Plant Cell and Animal Cell