Question

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

• A. $2{\log }_{3}e$
• B. $\frac{1}{2}{\log }_{3}e$
• C. ${\log }_{3}e$
• D. $2{\log }_{e}3$

Answer 1

The correct option is A $2{\log }_{3}e$

Using mean value theorem on the interval $[1,3]$

$\therefore f^{\prime }\left(c\right)=\frac{f\left(3\right)-f\left(1\right)}{3-1}$

Substituting values, we get

$\Rightarrow \frac{1}{c}=\frac{\log 3-\log 1}{2}$

$\Rightarrow c=\frac{2}{{\log }_{e}3}=2{\log }_{3}e$