How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [0,7] for #f(x)=1/((x+1)^6)#?

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Answer 1 · 2016-12-04T17:38:58

Solve #f'(x) = (f(7)-f(0))/(7-0)# on the interval #(0,7)#. I get #c = -1 + root(7)((6 * 7 * 8^7)/(8^6-1))#. Do as much arithmetic as you like.

#f'(x) = (-6)/(x+1)^7#

#f(7) = 1/8^6# and #f(0) = 1#

So we need to solve

#(-6)/(x+1)^7 = (1/8^6 - 1)/7#

I get #c = -1 + root(7)((6 * 7 * 8^7)/(8^6-1))#. Do as much arithmetic as you like.