Mathematics - Application of Derivative Question with Solution | TestHub

Using Lagrange's Mean Value Theorem Let $\mathrm{f}(\mathrm{x})$ be a function defined on $[\mathrm{a},\mathrm{b}]$ then, $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$ $\mathrm{c}\in [\mathrm{a},\mathrm{b}]$ $\therefore$ Given $f(x)={\log }_{e}x\therefore f^{\prime }(x)=\frac{1}{x}$ $\therefore$ equation (i) become $$\begin{array}{rl}& \frac{1}{c}=\frac{f(3)-f(1)}{3-1}\\ \Rightarrow & \frac{1}{c}=\frac{{\log }_{e}3-{\log }_{e}1}{2}=\frac{{\log }_{\mathrm{e}}3}{2}\Rightarrow c=\frac{2}{{\log }_{\mathrm{e}}3}\\ \Rightarrow & c=2{\log }_{3}e\end{array}$$