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

 

 

Option: 1

\log _3 e
 


Option: 2

\log _e 3
 


Option: 3

2 \log _3 e
 


Option: 4

\frac{1}{2} \log _e 3


Answers (1)

best_answer

 By LMVT,
\mathrm{\begin{aligned} & f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}=\frac{f(3)-f(1)}{3-1} \\ & \Rightarrow \quad f^{\prime}(c)=\frac{\log _e 3-\log _e 1}{2}=\frac{1}{2} \log _e 3 \\ & \Rightarrow \frac{1}{c}=\frac{1}{2} \log _e 3=\frac{1}{2 \log _3 e} \\ & \therefore c=2 \log _3 e . \end{aligned} }
 

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Rishi

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