(B) Given that $f:[1,3] \rightarrow R$ is continuous on $[1,3]$ and differentiable in $(1,3)$ with $f^{\prime}(x)=|f(x)|^{2}+4$. By applying the Lagra

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

(B) Given that $f:[1,3] \rightarrow R$ is continuous on $[1,3]$ and differentiable in $(1,3)$ with $f^{\prime}(x)=|f(x)|^{2}+4$. By applying the Lagrange Mean Value Theorem $(LMVT)$, there exists at least one point $c \in (1,3)$ such that: $\frac{f(3)-f(1)}{3-1} = f^{\prime}(c)$ $\frac{f(3)-f(1)}{2} = |f(c)|^{2} + 4$ Since $|f(c)|^{2} \geq 0$, we have $|f(c)|^{2} + 4 \geq 4$. Therefore, $\frac{f(3)-f(1)}{2} \geq 4$, which implies $f(3)-f(1) \geq 8$. Thus, the statement $f(3)-f(1)=5$ is false.

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

Medium

Let $f:[1,3] \rightarrow R$ be a continuous function that is differentiable in $(1,3)$ and $f^{\prime}(x)=|f(x)|^{2}+4$ for all $x \in(1,3).$ Then,

WBJEE 2019 All WBJEE

A
$f(3)-f(1)=5$ is true
B
$f(3)-f(1)=5$ is false
C
$f(3)-f(1)=7$ is true
D
$f(3)-f(1) < 0$ only at one point of $(1,3)$

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

Previous Year

WBJEE 2019 Paper All WBJEE years →

Consider the function $f(x) = \begin{cases} x \sin \frac{\pi}{x} & \text{for } x > 0 \\ 0 & \text{for } x = 0 \end{cases}$. Then the number of points in $(0, 1)$ where the derivative $f'(x)$ vanishes is:

View Solution

Values of $c$ as per Rolle's theorem for $f(x)=\sin x+\cos x+6$ on $[0, 2\pi]$ are

EasyMHT CET 2023

View Solution

Examine if Rolle's Theorem is applicable to the function $f(x) = [x]$ for $x \in [5, 9]$. Can you say something about the converse of Rolle's Theorem from this example?

Medium

View Solution

For the curve $y = x^3$ in the interval $[-2, 2]$,find the abscissae of the points where the slope of the tangent is equal to the slope of the secant line passing through the endpoints of the interval,as per the Mean Value Theorem.

MediumMHT CET 2025

View Solution

Let $f(x) = \begin{cases} x^\alpha \ln x, & x > 0 \\ 0, & x = 0 \end{cases}$. Rolle's theorem is applicable to $f$ for $x \in [0, 1]$ if $\alpha = $

DifficultIIT 2004

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

Let $f:[1,3] \rightarrow R$ be a continuous function that is differentiable in $(1,3)$ and $f^{\prime}(x)=|f(x)|^{2}+4$ for all $x \in(1,3).$ Then,

Similar Questions

Consider the function $f(x) = \begin{cases} x \sin \frac{\pi}{x} & \text{for } x > 0 \\ 0 & \text{for } x = 0 \end{cases}$. Then the number of points in $(0, 1)$ where the derivative $f'(x)$ vanishes is:

Values of $c$ as per Rolle's theorem for $f(x)=\sin x+\cos x+6$ on $[0, 2\pi]$ are

Examine if Rolle's Theorem is applicable to the function $f(x) = [x]$ for $x \in [5, 9]$. Can you say something about the converse of Rolle's Theorem from this example?

For the curve $y = x^3$ in the interval $[-2, 2]$,find the abscissae of the points where the slope of the tangent is equal to the slope of the secant line passing through the endpoints of the interval,as per the Mean Value Theorem.

Let $f(x) = \begin{cases} x^\alpha \ln x, & x > 0 \\ 0, & x = 0 \end{cases}$. Rolle's theorem is applicable to $f$ for $x \in [0, 1]$ if $\alpha = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module