Mean Value Theorem

1. Understand the problem:

We need to find the value of in the interval that satisfies the Mean Value Theorem (MVT) for the function on the interval .

2. Verify MVT conditions:

The function is a polynomial, so it is continuous on and differentiable on . Thus, MVT applies.

3. Compute and :

4. Compute the average rate of change:

5. Find the derivative :

First, expand :

Now, differentiate:

6. Solve for :

Solutions: or .

Since , the valid solution is .

7. Match the result to the options:

The value corresponds to option (B).

Correct Answer: (B)