#x^3+24x-16# [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:
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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.
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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.
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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 ).
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Calculate the derivative of ( f(x) ) with respect to ( x ), ( f'(x) ), which is ( 3x^2 + 24 ).
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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} ]
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Once you find the value of ( c ), verify that it lies within the interval ((0,4)).
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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 one-sided 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 one-sided 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 one-sided 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 one-sided 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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