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Problem 8:
Let on the interval .
. Apply the MVT to find the value(s) of in where .

Ask by Lane Dawson. in the United States
Jan 23,2025

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Answer

The values of in that satisfy the Mean Value Theorem are and .

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The Deep Dive

To find the values of using the Mean Value Theorem (MVT), we begin by calculating and .
First, we evaluate:
Next, we calculate :
Now, we need to find and set it equal to 8:
Calculating the derivative:
Setting :
Now, we will use the quadratic formula, :
Calculating the approximate values of these roots, we find:
Now, evaluating , we calculate:
Since needs to be in the interval , both and are valid solutions. Thus, the values of are approximately:

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