A value of $$c$$ for which conclusion of Mean Value Theorem holds for the function $$f\left( x \right) = {\log _e}x$$ on the interval $$\left[ {1,3} \right]$$ is
A
$${\log _3}e$$
B
$${\log _e}3$$
C
$$2\,\,{\log _3}e$$
D
$${1 \over 2}{\log _3}e$$
Explanation
Using Lagrange's Mean Value Theorem
Let $$f(x)$$ be a function defined on $$\left[ {a,b} \right]$$
then, $$f'\left( c \right) = {{f\left( b \right) - f\left( a \right)} \over {b - a}}\,\,\,\,\,\,\,\,\,\,\,\,....\left( i \right)$$
$$c\,\, \in \left[ {a,b} \right]$$
$$\therefore$$ Given $$f\left( x \right) = {\log _e}x$$
$$\therefore$$ $$f'\left( x \right) = {1 \over x}$$