#x^3+24x16# [0,4] verify mean value theorem?
We seek to verify the Mean Value Theorem for the function
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To verify the Mean Value Theorem for the function ( f(x) = x^3 + 24x  16 ) over the interval ([0,4]), follow these steps:

Verify that the function ( f(x) ) is continuous on the closed interval ([0,4]). This function is a polynomial function, and polynomial functions are continuous for all real numbers.

Verify that the function ( f(x) ) is differentiable on the open interval ((0,4)). Since ( f(x) ) is a polynomial function, it is differentiable for all real numbers.

Calculate the average rate of change of ( f(x) ) over the interval ([0,4]) using the formula: [ \frac{f(b)  f(a)}{b  a} ] where ( a = 0 ) and ( b = 4 ).

Calculate the derivative of ( f(x) ) with respect to ( x ), ( f'(x) ), which is ( 3x^2 + 24 ).

Find the value of ( c ) in the open interval ((0,4)) such that ( f'(c) ) is equal to the average rate of change of ( f(x) ) over ([0,4]), by solving the equation: [ f'(c) = \frac{f(4)  f(0)}{4  0} ]

Once you find the value of ( c ), verify that it lies within the interval ((0,4)).

If ( c ) lies within the interval ((0,4)), then the Mean Value Theorem is verified for the function ( f(x) ) over the interval ([0,4]), and the theorem states that there exists at least one ( c ) such that ( f'(c) = \frac{f(4)  f(0)}{4  0} ).
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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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