How do you determine all values of c that satisfy the mean value theorem on the interval [3,2] for #f(x) = x^2 + 4x + 2#?
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The conclusion of the mean value theorem says:
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To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ([3, 2]) for ( f(x) = x^2 + 4x + 2 ), follow these steps:

Find the average rate of change of ( f(x) ) on the interval ([3, 2]). This is calculated as ( \frac{f(2)  f(3)}{2  (3)} ).

Compute ( f(2) ) and ( f(3) ) using the given function ( f(x) ).

Set the average rate of change equal to the derivative of ( f(x) ) evaluated at ( c ). This is expressed as ( f'(c) = \frac{f(2)  f(3)}{2  (3)} ).

Calculate ( f'(x) ), the derivative of ( f(x) ), which is ( f'(x) = 2x + 4 ).

Solve the equation ( f'(c) = \frac{f(2)  f(3)}{2  (3)} ) for ( c ).

Substitute the values of ( f(2) ) and ( f(3) ) into the equation and solve for ( c ).

The value(s) of ( c ) obtained are the value(s) that satisfy the Mean Value Theorem on the interval ([3, 2]) for the given function ( f(x) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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