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

Answer 1

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

$f\left(x\right)=lo{g}_{e}x$

$f^{\prime }\left(x\right)=\frac{1}{x}$

By Lagrange's mean value theorem,for c in (1,3)

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

$\Rightarrow \frac{1}{c}=\frac{lo{g}_{e}3}{2}$

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

$\Rightarrow c=2lo{g}_{3}e$