(C) Given that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(a)=0$ and $f(b)=0$. By Rolle's theorem, there exists at least one

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$f^{\prime \prime}(x) = 0$ for some $x \in(a, b)$

(C) Given that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(a)=0$ and $f(b)=0$. By Rolle's theorem, there exists at least one $c \in (a, b)$ such that $f^{\prime}(c)=0$. Since $f^{\prime}(a)=0$ and $f^{\prime}(c)=0$, and $f^{\prime}$ is continuous on $[a, c]$ and differentiable on $(a, c)$, we can apply Rolle's theorem to $f^{\prime}$ on the interval $[a, c]$. Therefore, there exists at least one $k \in (a, c)$ such that $f^{\prime \prime}(k)=0$. Since $(a, c) \subset (a, b)$, there exists $x \in (a, b)$ such that $f^{\prime \prime}(x)=0$.

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

Easy

Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then:

WBJEE 2015 All WBJEE

A
$f^{\prime \prime}(a)=0$
B
$f^{\prime}(x)=0$ for some $x \in(a, b)$
C
$f^{\prime \prime}(x) = 0$ for some $x \in(a, b)$
D
$f^{\prime \prime \prime}(x)=0$ for some $x \in(a, b)$

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Class 12

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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Consider all functions given in List-$I$ in the interval $[1,3]$. List-$II$ has the values of '$c$' obtained by applying Lagrange's Mean Value Theorem $(LMVT)$ on the functions of List-$I$. Match the functions and values of '$c$'.

List-$I$ List-$II$

$A. |x-1|$ $I. 2 \log (e^3+e^2)$

$B. \log x$ $II. 2$

$C. x^2+x+1$ $III. \log_3 e^2$

$D. e^x$ $IV. \sqrt{2}$

$V. \log \left(\frac{e^3-e}{2}\right)$

List-$I$	List-$II$
$A. \|x-1\|$	$I. 2 \log (e^3+e^2)$
$B. \log x$	$II. 2$
$C. x^2+x+1$	$III. \log_3 e^2$
$D. e^x$	$IV. \sqrt{2}$
	$V. \log \left(\frac{e^3-e}{2}\right)$

DifficultTS EAMCET 2025

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If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

Medium

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Let $f$ be a function which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$, then which of the following is true?

MediumWBJEE 2025

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If the function $f(x) = x(x+3)e^{-x/2}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then a root of $f'(x) = 0$ is

EasyTS EAMCET 2021

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If the function $f(x) = ax^3 + bx^2 + 11x - 6$ satisfies the conditions of Rolle's theorem for the interval $[1, 3]$ and $f'\left( 2 + \frac{1}{\sqrt{3}} \right) = 0$,then the values of $a$ and $b$ are respectively

Medium

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Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then:

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If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

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