How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function #f(x) = x^2 + 4x + 2# on the interval [3,2]?
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To find the number ( c ) that satisfies the conclusion of the Mean Value Theorem for the function ( f(x) = x^2 + 4x + 2 ) on the interval ([3, 2]), follow these steps:

Calculate ( f(3) ) and ( f(2) ):
( f(3) = (3)^2 + 4(3) + 2 = 9  12 + 2 = 1 )
( f(2) = (2)^2 + 4(2) + 2 = 4  8 + 2 = 2 )

Compute the average rate of change of ( f(x) ) over the interval ([3, 2]):
[ \text{Average rate of change} = \frac{f(2)  f(3)}{(2)  (3)} ]
[ \text{Average rate of change} = \frac{2  (1)}{2 + 3} = \frac{3}{1} = 3 ]

Find the derivative ( f'(x) ):
( f'(x) = \frac{d}{dx} (x^2 + 4x + 2) = 2x + 4 )

Set up the Mean Value Theorem equation:
[ f' (c) = \text{Average rate of change} ]
[ 2c + 4 = 3 ]

Solve for ( c ):
[ 2c = 3  4 ]
[ 2c = 7 ]
[ c = \frac{7}{2} ]
So, the number ( c ) that satisfies the conclusion of the Mean Value Theorem for the function ( f(x) = x^2 + 4x + 2 ) on the interval ([3, 2]) is ( c = \frac{7}{2} ) or ( c = 3.5 ).
By signing up, you agree to our Terms of Service and Privacy Policy
To find the number ( c ) that satisfies the conclusion of 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]). This is done by finding the difference in ( f(x) ) at the endpoints of the interval and dividing by the difference in ( x ) values. [ \text{Average rate of change} = \frac{f(2)  f(3)}{2  (3)} ]

Find the values of ( f(2) ) and ( f(3) ) by plugging ( x = 2 ) and ( x = 3 ) into the function ( f(x) = x^2 + 4x + 2 ). [ f(2) = (2)^2 + 4(2) + 2 ] [ f(3) = (3)^2 + 4(3) + 2 ]

Calculate the difference between ( f(2) ) and ( f(3) ). [ f(2)  f(3) ]

Calculate ( c ) using the Mean Value Theorem formula, which states that there exists a number ( c ) in the interval ([3, 2]) such that the instantaneous rate of change (derivative) at ( c ) equals the average rate of change over the interval. [ f'(c) = \frac{f(2)  f(3)}{2  (3)} ]

Find the derivative ( f'(x) ) by differentiating ( f(x) ) with respect to ( x ). [ f'(x) = 2x + 4 ]

Set up the equation ( f'(c) = \frac{f(2)  f(3)}{2  (3)} ) and solve for ( c ). [ 2c + 4 = \frac{f(2)  f(3)}{2  (3)} ]

Substitute the values of ( f(2) ) and ( f(3) ) into the equation and solve for ( c ).
After completing these steps, you will find the value of ( c ) that satisfies the conclusion of the Mean Value Theorem for the given function and interval.
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