Question

For the function $f\left(x\right)=e^x,a=0,b=1$, the value of $c$ in mean value theorem will be

• A. $\log x$
• B. $\log (e–1)$
• C. $0$
• D. $1$

Answer 1

The correct option is B

$\log (e–1)$

Explanation for the correct option:

Step 1. Mean value theorem.

Formula, for any given function M.V.T is given by,$\mathit{f}\mathbf{\text{'}}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{f}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{a}\mathbf{\right)}}{\mathbf{b}\mathbf{-}\mathbf{a}}\mathbf{}$

Step 2. Finding the value of $c$

Using the above formula,

$\begin{array}{rcl}Givenf\left(x\right)& =& e^{x}anda=0,b=1\\ f\text{'}\left(c\right)& =& \frac{f\left(b\right)-f\left(a\right)}{b-a}\\ f\text{'}\left(c\right)& =& \frac{f\left(1\right)-f\left(0\right)}{1-0}\\ e^c& =& e-1\\ & & Taking\log bothside\\ c& =& \log (e–1)\end{array}$

Hence, the correct option is $\left(B\right)$ .