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.

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)f(x)=1/((x+1)^6)?