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) = 1 / (x-1)#; [2, 5]?

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Answer 1 · 2016-11-15T17:52:07

Solve #f'(x) = (f(5)-f(2))/(5-2)# on #(2,5)#

There is no "principle of the mean value theorem" to use here.

Use the rules for finding derivatives to get #f'(x) = (-1)/(x-1)^2#

Now use algebra to solve

#(-1)/(x-1)^2 = (f(5)-f(2))/(5-2)#

#(-1)/(x-1)^2 = (1/4-1)/3 = -1/4#

Thus #(x-1)^2 = 4#

So #x-1=2# and #x=3# #" "# OR #" "# #x-1=-2# and #x = -1#.

The only number that satisfies the conclusion of MVT is #3#.