Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. ,
E)None of these

Question

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval.  , 
E)None of these

Quizplus · Accepted Answer

By the Mean Value Theorem, we know that there exists a number $c$ in the interval $(11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11 , 11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11)$ such that $$f'(c)=\frac{f(11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11)-f(11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11)}{11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11-11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11}.$$ We have that $f(x)=\ln x$ and so $f'(x)=\frac{1}{x}$. Plugging this into the equation above we get: $$\frac{1}{c}=\frac{\ln(11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11)-\ln(11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11)}{11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11-11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11}.$$ Simplifying the right side gives: $$\frac{1}{c}=\frac{\ln\left(\frac{11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11}{11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11}ight)}{11eaa3e5_55d7_2703_9f8f_dc4809b1d062_TB5971_11}.$$ The fraction in the logarithm on the right is equal to $\frac{11eaa3e5_55d7_686d_9f8f_69264e6485e0_TB5971_11}{11eaa3e5_55d7_415c_9f8f_89d944caa6e3_TB5971_11}$ and simplifies to $11eaa3e5_55d7_8f81_9f8f_83b5585437df_TB5971_11$. Then we have $$\frac{1}{c}=\frac{\ln(11eaa3e5_55d7_8f81_9f8f_83b5585437df_TB5971_11)}{11eaa3e5_55d7_2703_9f8f_dc4809b1d062_TB5971_11}.$$ Solving for $c$ gives $c=11eaa3e5_55d7_8f81_9f8f_83b5585437df_TB5971_11$. Therefore, the answer is $\boxed{	extbf{(D)}}$.

Find the Number C That Satisfies the Conclusion of the Mean