# How do you verify that the function #f(x)=x/(x+6)# satisfies the hypotheses of The Mean Value Theorem on the given interval [0,1] and then find the number c that satisfy the conclusion of The Mean Value Theorem?

So we find them out:

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To verify that the function ( f(x) = \frac{x}{x + 6} ) satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval ([0,1]), you need to check two conditions:

- ( f(x) ) is continuous on ([0,1]).
- ( f(x) ) is differentiable on ((0,1)).

First, let's check if ( f(x) ) is continuous on ([0,1]). The function ( f(x) ) is a rational function, and it is continuous everywhere in its domain except where its denominator ( x + 6 ) equals zero. However, on the interval ([0,1]), ( x + 6 ) is never zero, so ( f(x) ) is continuous on ([0,1]).

Next, let's check if ( f(x) ) is differentiable on ((0,1)). ( f(x) ) is differentiable everywhere in its domain except where it's not defined or where its derivative doesn't exist. Since ( f(x) ) is defined and continuous on ([0,1]), it is also differentiable on ((0,1)).

Since both conditions are satisfied, we can apply the Mean Value Theorem. According to MVT, there exists a number ( c ) in ((0,1)) such that ( f'(c) ) equals the average rate of change of ( f(x) ) over the interval ([0,1]).

To find ( c ), first, compute ( f'(x) ). Then, find the average rate of change of ( f(x) ) over ([0,1]). Finally, solve ( f'(c) = ) average rate of change for ( c ).

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