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)#?

Answer 1

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:

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

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

  3. Set the average rate of change equal to the derivative ( f'(c) ) and solve for ( c ). ( \frac{f(7) - f(0)}{7-0} = 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} )

  4. 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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Answer 2

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.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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