Examine the Rolles theorem is applicable to the followng function. Find the number of points the following function is not continous?
$f (x) = [x]$ for $x$ $ϵ$ $[2, 2]$

Open in App

Solution

Rolle's Theorem holds for a function $f : [a, b] \to R$ , if following three conditions holds
(i) $f$ is continuous on $[a, b]$
(ii) $f$ is differentiable on $(a, b)$
(iii) $f (a) = f (b)$
Then, there exists some $c \in (a, b)$ such that $f^{'} (c) = 0$
So, Rolle's Theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis.

Given function $f (x) = [x]$ for $x \in [- 2, 2]$
Since, the greatest integer function is not continuous at integral points.
So, $f (x)$ is not continuous at $x = - 2, - 1, 0, 1, 2$
$f (x)$ is not continuous in $[- 2, 2]$ .

Suggest Corrections

Similar questions

Examine the applicability of Mean Value Theorem for the following function.

f (x) = [x

] for

x

ϵ

[2, 2]

Q. Examine if Rolles theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolles theorem from these example ?
(i)

f (x) = [x]

for

x \in [5, 9]

(ii)

f (x) = [x]

for

x \in [- 2, 2]

(iii)

f (x) = x

^{2} - 1

for

x \in [1, 2]

Examine if Rolle's theorem is applicable to any of the following functions. Can you say something about the converse of Rolle's theorem from these example ?

(i) $f (x) = [x] f o r x ϵ [- 2, 2]$

Q. Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?

Examine if Rolle's theorem is applicable to any of the following functions. Can you say something about the converse of Rolle's theorem from these example ?

(i) $f (x) = [x] f o r x ϵ [5, 9]$

(i) $f (x) = [x] f o r x ϵ [- 2, 2]$

(iii) $f (x) = x^{2} - 1 f o r x ϵ [5, 9]$ .

Join BYJU'S Learning Program

Examine the Rolles theorem is applicable to the followng function. Find the number of points the following function is not continous?f(x)=[x] for x ϵ [2,2]

Examine the Rolles theorem is applicable to the followng function. Find the number of points the following function is not continous?
$f (x) = [x]$ for $x$ $ϵ$ $[2, 2]$