Q60 / 120 · JEE Main 2007 · Mathematics

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f(x)=$ $\log _{\mathrm{e}} x$ on the interval $[1,3]$ is

Correct answer
A
$2 \log _3 e$
B
$\frac{1}{2} \log _{\mathrm{e}} 3$
C
$\log _3 e$
D
$\log _e 3$

Solution

Using mean value theorem $\begin{aligned} & f^{\prime}(c)=\frac{f(3)-f(1)}{3-1} \\ & \Rightarrow \frac{1}{c}=\frac{\log 3-\log 1}{2} \\ & \Rightarrow c=\frac{2}{\log _e 3}=2 \log _3 e. \end{aligned}$

Hence, the correct answer is option: 'A'.

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