Question

If we apply mean value theorem for $f\left(x\right)=\log xx\in [1,2]$, then the value of $c$ is

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

Answer 1

Answer: (C)

${\log }_{2}e$

$f\left(x\right)-logx,x\in [1,2]$

Using mean value Theorem,

$\frac{f\left(b\right)-f\left(a\right)}{(b-a)}=f^{\prime }\left(c\right)\Rightarrow \frac{f\left(2\right)-f\left(1\right)}{(2-1)}=f^{\prime }\left(c\right)f^{\prime }\left(x\right)=\frac{1}{x}\Rightarrow f^{\prime }\left(c\right)=\frac{1}{c}\Rightarrow \frac{log\left(2\right)-log\left(1\right)}{1}=\frac{1}{c}\Rightarrow log\left(\frac{1}{2}\right)=\frac{1}{c}\Rightarrow c\Rightarrow \frac{1}{log2}={\log }_2\left(e\right)\Rightarrow c={\log }_2\left(e\right)$