Answer
Verified
377.4k+ 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
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE