(B) Let $f(t) = \log t$. By Lagrange's Mean Value Theorem on the interval $[e, x]$,there exists a $c \in (e, x)$ such that $\frac{f(x) - f(e)}{x - e}

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(B) Let $f(t) = \log t$. By Lagrange's Mean Value Theorem on the interval $[e, x]$,there exists a $c \in (e, x)$ such that $\frac{f(x) - f(e)}{x - e} = f'(c)$.Since $f'(t) = \frac{1}{t}$,we have $\frac{\log x - \log e}{x - e} = \frac{1}{c}$.Given $e Since $e \approx 2.718$,$\frac{1}{e} \approx 0.367$.Thus,$0 The greatest integer function $[y]$ for $0 Therefore,the value is $0$.

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

The value of $\left[ \frac{\log (x/e)}{x - e} \right]$ for all $x > e$ is equal to (where $[.]$ denotes the greatest integer function).

A
$1$
B
$0$
C
$2$
D
does not take a unique value

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

Consider the function $f(x) = e^{-2x} \sin 2x$ over the interval $(0, \pi/2)$. $A$ real number $c \in (0, \pi/2)$,as guaranteed by Rolle's theorem,such that $f'(c) = 0$ is

Easy

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Let $f(x)=(x-1)(x-2)(x-3)$,where $x \in [0,4]$. Find the values of $c$ if Lagrange's Mean Value Theorem $(LMVT)$ can be applied.

MediumMHT CET 2024

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If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

Medium

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For the function $f(x) = \log(\sin x)$ in the interval $[\frac{\pi}{6}, \frac{5\pi}{6}]$,what is the value of $c$ according to Lagrange's Mean Value Theorem?

Easy

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In the interval $[0, 1]$,Lagrange's Mean Value Theorem is $NOT$ applicable to which of the following functions?

MediumIIT 2003

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The value of $\left[ \frac{\log (x/e)}{x - e} \right]$ for all $x > e$ is equal to (where $[.]$ denotes the greatest integer function).

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Consider the function $f(x) = e^{-2x} \sin 2x$ over the interval $(0, \pi/2)$. $A$ real number $c \in (0, \pi/2)$,as guaranteed by Rolle's theorem,such that $f'(c) = 0$ is

Let $f(x)=(x-1)(x-2)(x-3)$,where $x \in [0,4]$. Find the values of $c$ if Lagrange's Mean Value Theorem $(LMVT)$ can be applied.

If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

For the function $f(x) = \log(\sin x)$ in the interval $[\frac{\pi}{6}, \frac{5\pi}{6}]$,what is the value of $c$ according to Lagrange's Mean Value Theorem?

In the interval $[0, 1]$,Lagrange's Mean Value Theorem is $NOT$ applicable to which of the following functions?

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