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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Answer 1 · 2016-09-19T17:26:17

See below.

What does the conclusion of the Mean Value Theorem say?

There is a #c# in #(2,6)# such that

#f'(c) = (f(6)-f(2))/(6-2)#

To find the #c#. Find #f'(x)#, find #(f(6)-f(2))/(6-2)#.

Set them equal to each other and solve the equation. Any (and all) solutions in #(2,6)# are values for #c#.

Solve

#1/(x-1) = ln5/4#

#x-1 = 4/ln5#

#x=(4+ln5)/ln5 ~~ 3.485#