(A) By the Lagrange's Mean Value Theorem $(LMVT)$,there exists a $c \in (1, 6)$ such that $f'(c) = \frac{f(6) - f(1)}{6 - 1}$.Given $f'(x) \geq 2$ for

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

(A) By the Lagrange's Mean Value Theorem $(LMVT)$,there exists a $c \in (1, 6)$ such that $f'(c) = \frac{f(6) - f(1)}{6 - 1}$.Given $f'(x) \geq 2$ for all $x \in [1, 6]$,we have $f'(c) \geq 2$.Substituting the values,we get $\frac{f(6) - (-2)}{5} \geq 2$.$f(6) + 2 \geq 10$.$f(6) \geq 8$.

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

Medium

Suppose $f$ is differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for all $x \in [1, 6]$,then:

A
$f(6) \geq 8$
B
$f(6) < 8$
C
$f(6) < 5$
D
$f(6) = 5$

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

The function $f(x) = x(x + 3)e^{-(1/2)x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$. The value of $c$ is

Medium

View Solution

If $f(x) = \log(\sin x)$,$x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$,then the value of $c$ by applying Lagrange's Mean Value Theorem $(LMVT)$ is:

EasyMHT CET 2020

View Solution

Let $f(x)$ satisfy the requirements of Lagrange's Mean Value Theorem in $[0, 2]$. If $f(0) = 0$ and $|f'(x)| \leqslant \frac{1}{2}$ for all $x \in [0, 2]$,then-

View Solution

$A$ value of $c$ for which the conclusion of the Mean Value Theorem holds for the function $f(x) = \log_e x$ on the interval $[1, 3]$ is

MediumAIEEE 2007

View Solution

Verify Rolle's theorem for the function $y=x^{2}+2$ on the interval $[-2, 2]$.

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

Suppose $f$ is differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for all $x \in [1, 6]$,then:

Similar Questions

The function $f(x) = x(x + 3)e^{-(1/2)x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$. The value of $c$ is

If $f(x) = \log(\sin x)$,$x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$,then the value of $c$ by applying Lagrange's Mean Value Theorem $(LMVT)$ is:

Let $f(x)$ satisfy the requirements of Lagrange's Mean Value Theorem in $[0, 2]$. If $f(0) = 0$ and $|f'(x)| \leqslant \frac{1}{2}$ for all $x \in [0, 2]$,then-

$A$ value of $c$ for which the conclusion of the Mean Value Theorem holds for the function $f(x) = \log_e x$ on the interval $[1, 3]$ is

Verify Rolle's theorem for the function $y=x^{2}+2$ on the interval $[-2, 2]$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module