# How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [0, pi] for #f(x)=2sin(x)+sin(4x) #?

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To determine all values of ( c ) that satisfy the conclusion of the Mean Value Theorem on the interval ([0, \pi]) for ( f(x) = 2\sin(x) + \sin(4x) ), you first need to find the average rate of change of the function over the interval ([0, \pi]). This is given by:

[ f'_{\text{avg}} = \frac{f(\pi) - f(0)}{\pi - 0} ]

Next, find the derivative of ( f(x) ) with respect to ( x ), ( f'(x) ), and then find where ( f'(x) = f'*{\text{avg}} ). The values of ( c ) for which ( f'(c) = f'*{\text{avg}} ) will satisfy the conclusion of the Mean Value Theorem.

So, compute ( f'(x) ) and ( f'*{\text{avg}} ), then solve ( f'(x) = f'*{\text{avg}} ) for ( x ) to find ( c ) values.

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