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Examine if Rolle’s Theorem is applicable to following function:
$f\left( x \right)=\left[ x \right]$ for$x\in \left[ 5,9 \right]$
Can you say something about the converse of Rolle’s Theorem from these functions?

Last updated date: 23rd Jul 2024
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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.