Question

A value of c for which the conclusion of mean value Theorem holds for the function f(x) = log_e x on the interval [1, 3] is
1. 2 log₃ e
2. \(\frac{1}{2}{\log _3}e\)
3. log₃ e
4. log_e 3

Answer 1

Correct Answer - Option 1 : 2 log₃ e

Concept:

For mean value theorem in [a, b]

\(f'\left( c \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\)

Calculation:

f(x) = log_e x

\(f'\left( x \right) = \frac{1}{x}\;\therefore f'\left( c \right) = \frac{1}{c}\)

\(\frac{1}{c} = \frac{{f\left( 3 \right) - f\left( 1 \right)}}{{3 - 1}} = \frac{{{{\log }_e}3 - {{\log }_e}1}}{{3 - 1}}\)

\(\frac{1}{c} = \frac{{{{\log }_e}3}}{2}\)

\(c = \frac{2}{{{{\log }_e}3}} = 2{\log _3}e\)