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)#?
To determine all values of ( c ) that satisfy the conclusion of the Mean Value Theorem on the interval ([0,7]) for ( f(x) = \frac{1}{(x+1)^6} ), follow these steps:

Find the derivative of ( f(x) ), denoted as ( f'(x) ). ( f'(x) = 6(x+1)^{7} )

Determine the average rate of change of ( f(x) ) over the interval ([0,7]). ( \text{Average rate of change} = \frac{f(7)  f(0)}{70} )

Set the average rate of change equal to the derivative ( f'(c) ) and solve for ( c ). ( \frac{f(7)  f(0)}{70} = f'(c) ) ( \frac{\frac{1}{8}  1}{7} = 6(c+1)^{7} ) ( \frac{7}{56} = 6(c+1)^{7} ) ( \frac{1}{8} = (c+1)^{7} )

Solve for ( c ) using the equation ( \frac{1}{8} = (c+1)^{7} ). ( (c+1)^{7} = \frac{1}{8} ) ( (c+1) = \sqrt[7]{\frac{1}{8}} ) ( c+1 = \frac{1}{2} ) ( c = \frac{1}{2}  1 ) ( c = \frac{1}{2} )
Therefore, the value of ( c ) that satisfies the conclusion of the Mean Value Theorem on the interval ([0,7]) for ( f(x) = \frac{1}{(x+1)^6} ) is ( c = \frac{1}{2} ).
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