Question

If LMVT holds for the function $f\left(x\right)=\ln x$ on the interval $[1,3]$ at $x=c,$ then which of the following is not true

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

Answer 1

The correct option is C $c=\frac{1}{2}{\log }_{3}e$

$f\left(x\right)=\ln x$, $x\in [1,3]$
$f\left(x\right)$ is continuous in $[1,3]$ and differentiable in $(1,3)$
$\therefore f^{\prime }\left(c\right)=\frac{f\left(3\right)-f\left(1\right)}{3-1}$
$\Rightarrow \frac{1}{c}=\frac{\ln 3}{2}$
$\Rightarrow c=\frac{2}{\ln 3}\in (1,3)$
$\Rightarrow c=2{\log }_{3}e$
$\Rightarrow c={\log }_3{e}^2\Rightarrow 3^c=e^2$
So $3^c=7.4$ (approx.)
$\Rightarrow 1<c<2$