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  4. A value of c for which the conclusion of Mean Value Theorem holds for the function f(x) = logex on the interval [1, 3] is

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Q. A value of $c$ for which the conclusion of Mean Value Theorem holds for the function $f(x) = log_ex$ on the interval $[1, 3]$ is

AIEEEAIEEE 2007Continuity and Differentiability

Solution:

Using mean value theorem
$f '\left(c\right)=\frac{f \left(3\right)-f \left(1\right)}{3-1}$
$\Rightarrow \frac{1}{c}=\frac{log\,3-log\,1}{2}$
$\Rightarrow c=\frac{2}{log_{e}\,3}=2log_{e}\,e.$