# Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem #f(x) = ln (x-1)#; [2, 6]?

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What does the conclusion of the Mean Value Theorem say?

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To find all numbers ( c ) that satisfy the conclusion of the Mean Value Theorem for the function ( f(x) = \ln(x-1) ) on the interval ( [2, 6] ), follow these steps:

- Verify that the function ( f(x) ) is continuous on the closed interval ( [2, 6] ) and differentiable on the open interval ( (2, 6) ).
- Calculate the derivative of ( f(x) ) with respect to ( x ). The derivative of ( \ln(x-1) ) is ( \frac{1}{x-1} ).
- Apply the Mean Value Theorem, which states that there exists a number ( c ) in the open interval ( (2, 6) ) such that ( f'(c) ) equals the average rate of change of ( f(x) ) over ( [2, 6] ).
- Find the average rate of change of ( f(x) ) over ( [2, 6] ) by calculating ( \frac{f(6) - f(2)}{6 - 2} ).
- Set the derivative ( f'(c) ) equal to the average rate of change found in step 4 and solve for ( c ). The equation to solve is ( \frac{1}{c-1} = \frac{\ln(5) - \ln(1)}{6 - 2} ).
- Solve the equation for ( c ) to find all values of ( c ) that satisfy the conclusion of the Mean Value Theorem on the interval ( [2, 6] ).

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