(D) The Mean Value Theorem $(MVT)$ requires a function to be continuous on $[a, b]$ and differentiable on $(a, b)$.For option $D$,$f(x) = |x|$ is not

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(D) The Mean Value Theorem $(MVT)$ requires a function to be continuous on $[a, b]$ and differentiable on $(a, b)$.For option $D$,$f(x) = |x|$ is not differentiable at $x = 0$. Since $0 \in [0, 1]$,the function $f(x) = |x|$ is not differentiable on the interval $(0, 1)$.Therefore,the Mean Value Theorem cannot be applied to $f(x) = |x|$ on $[0, 1]$ because it fails the differentiability condition at $x = 0$.

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

Medium

According to the Mean Value Theorem,which of the following functions does not satisfy the conditions on the interval $[0, 1]$?

A
$f(x) = \begin{cases} \frac{1}{2} - x, & x < \frac{1}{2} \\ (\frac{1}{2} - x)^2, & x \ge \frac{1}{2} \end{cases}$
B
$f(x) = \begin{cases} \frac{\sin x}{x}, & x \ne 0 \\ 1, & x = 0 \end{cases}$
C
$f(x) = x |x|$
D
$f(x) = |x|$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

Continuity and Differentiation

Subtopic

Rolle’s theorem, Lagrange's mean value theorem

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

View Solution

If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:
$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.
$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.
$(c)$ Only $g(x)$ satisfies Rolle's Theorem.
Which of these statements is correct?

MediumKCET 2014

View Solution

The number of admissible values of $C$ obtained when the Lagrange's mean value theorem is applied for the function $f(x)=x$ on $[2,5]$ is

EasyTS EAMCET 2020

View Solution

For all twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R},$ with $f(0)=f(1)=f^{\prime}(0)=0,$ which of the following is true?

DifficultJEE MAIN 2020

View Solution

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f(6)$ is:

Medium

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

According to the Mean Value Theorem,which of the following functions does not satisfy the conditions on the interval $[0, 1]$?

Similar Questions

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

If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.$(c)$ Only $g(x)$ satisfies Rolle's Theorem.Which of these statements is correct?

The number of admissible values of $C$ obtained when the Lagrange's mean value theorem is applied for the function $f(x)=x$ on $[2,5]$ is

For all twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R},$ with $f(0)=f(1)=f^{\prime}(0)=0,$ which of the following is true?

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f(6)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:
$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.
$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.
$(c)$ Only $g(x)$ satisfies Rolle's Theorem.
Which of these statements is correct?