How do you find the value of c guaranteed by the mean value theorem if it can be applied for #f(x) = x^2 + 4x + 2# on the interval [3,2]?
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To find the value of ( c ) guaranteed by the Mean Value Theorem for the function ( f(x) = x^2 + 4x + 2 ) on the interval ([3, 2]), follow these steps:

Calculate the average rate of change of ( f(x) ) over the interval ([3, 2]) using the formula: [ \text{Average rate of change} = \frac{f(2)  f(3)}{2  (3)} ]

Evaluate ( f(2) ) and ( f(3) ) by substituting ( x = 2 ) and ( x = 3 ) into the function ( f(x) = x^2 + 4x + 2 ).

Once you have the average rate of change, set it equal to the derivative of ( f(x) ) with respect to ( x ) evaluated at some point ( c ) within the interval ([3, 2]): [ f'(c) = \text{Average rate of change} ]

Find the derivative ( f'(x) ) of ( f(x) ), then solve the equation ( f'(c) = \text{Average rate of change} ) for ( c ).

Once you find ( c ), verify that it lies within the interval ([3, 2]).

The value of ( c ) thus obtained is guaranteed by the Mean Value Theorem.
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