# How do you find all values of c that satisfy the Mean Value Theorem for integrals for #f(x)=x^4# on the interval [-4,4]?

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To find the value of ( c ) that satisfies the Mean Value Theorem for integrals for ( f(x) = x^4 ) on the interval ([-4, 4]), follow these steps:

- Evaluate the definite integral of ( f(x) ) over the interval ([-4, 4]):

[ \int_{-4}^{4} x^4 , dx ]

[ = \left[ \frac{x^5}{5} \right]_{-4}^{4} ]

[ = \frac{4^5}{5} - \frac{(-4)^5}{5} ]

[ = \frac{1024}{5} - \frac{-1024}{5} ]

[ = \frac{2048}{5} ]

[ = 409.6 ]

- Calculate the average value of ( f(x) ) over the interval ([-4, 4]):

[ \text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) , dx ]

[ = \frac{1}{4 - (-4)} \times 409.6 ]

[ = \frac{1}{8} \times 409.6 ]

[ = 51.2 ]

- Apply the Mean Value Theorem for integrals:

[ f(c) = \text{Average value of } f(x) ]

[ c^4 = 51.2 ]

- Solve for ( c ):

[ c^4 = 51.2 ]

[ c = \sqrt[4]{51.2} ]

[ c \approx 3.25 ]

Thus, the value of ( c ) that satisfies the Mean Value Theorem for integrals for ( f(x) = x^4 ) on the interval ([-4, 4]) is approximately ( c \approx 3.25 ).

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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.

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