Question

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

Answer 1

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

acccording to mean value theorm there exists a point c belongs to $(1,3)$ such that

$f^1\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$\frac{1}{x}=\frac{log3-log1}{2}$

$c=\frac{2}{log3}$

$c=\frac{{log}^{e^2}}{log3}$

$c=lo{g}_3^{e^2}$

$c=2lo{g}_3^e$