Limits Continuity And Differentiability Question 15
Question: 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
Options:
A) $ log_3e $
B) $ log _{e}3 $
C) $ 2log_3e $
D) $ \frac{1}{2}{\log_3}e $
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Answer:
Correct Answer: C
Solution:
Using Lagrange’s Mean Value Theorem
Let f(x) be a function defined on [a, b] then, $ f’(c)=\frac{f(b)-f(a)}{b-a}…(i) $
$ c\in [a,b]\therefore Givenf(x)={\log _{e}}x\therefore f’(x)=\frac{1}{x} $
$ \therefore $ equation (i) become $ \frac{1}{c}=\frac{f(3)-f(1)}{3-1} $
$ \Rightarrow \frac{1}{c}=\frac{{\log _{e}}3-{\log _{e}}1}{2}=\frac{{\log _{e}}3}{2} $
$ \Rightarrow c=\frac{2}{{\log _{e}}3}\Rightarrow c=2{\log_3}e $
BETA
