(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$,continuous on $[a, b]$,and differentiable on $(a, b)$.Her

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Rolle's theorem is not applicable to $f$ in $D$

(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$,continuous on $[a, b]$,and differentiable on $(a, b)$.Here,the domain $D = [0, 1] \cup [2, 4]$ is not a single closed interval. It is a union of two disjoint closed intervals.For Rolle's theorem to be applicable,the domain must be a single connected closed interval $[a, b]$.Since the domain $D$ is disconnected,Rolle's theorem is not applicable to $f$ in $D$.

Class 12 Mathematics Continuity and Differentiation Rolle’s theorem, Lagrange's mean value theorem

Easy

Let $f: D \rightarrow R$ where $D=[0,1] \cup [2,4]$ be defined by $f(x)=\begin{cases} x, & \text{if } x \in [0,1] \\ 4-x, & \text{if } x \in [2,4] \end{cases}$. Then,

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A
Rolle's theorem is applicable to $f$ in $D$
B
Rolle's theorem is not applicable to $f$ in $D$
C
there exists $\xi \in D$ for which $f^{\prime}(\xi)=0$ but Rolle's theorem is not applicable
D
$f$ is not continuous in $D$

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Rolle’s theorem, Lagrange's mean value theorem

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Examine if Rolle's Theorem is applicable to the function $f(x) = x^{2} - 1$ for $x \in [1, 2]$. Can you say something about the converse of Rolle's Theorem from this example?

Medium

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Let $f:(a, b) \rightarrow R$ be a twice differentiable function such that $f(x) = \int_{a}^{x} g(t) \, dt$ for a differentiable function $g(x)$. If $f(x) = 0$ has exactly five distinct roots in $(a, b)$,then $g(x) g'(x) = 0$ has at least:

DifficultJEE MAIN 2021

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If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

Medium

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For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

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The value $c$ of Lagrange's mean value theorem for $f(x)=e^{x}+24$ in $[0,1]$ is

EasyAP EAMCET 2024

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Let $f: D \rightarrow R$ where $D=[0,1] \cup [2,4]$ be defined by $f(x)=\begin{cases} x, & \text{if } x \in [0,1] \\ 4-x, & \text{if } x \in [2,4] \end{cases}$. Then,

Similar Questions

Examine if Rolle's Theorem is applicable to the function $f(x) = x^{2} - 1$ for $x \in [1, 2]$. Can you say something about the converse of Rolle's Theorem from this example?

Let $f:(a, b) \rightarrow R$ be a twice differentiable function such that $f(x) = \int_{a}^{x} g(t) \, dt$ for a differentiable function $g(x)$. If $f(x) = 0$ has exactly five distinct roots in $(a, b)$,then $g(x) g'(x) = 0$ has at least:

If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

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