Determine whether the statement is true or false.

If *f* is differentiable and *f*(−6) = *f*(6), then there is a number *c* such that |*c*| < 6 and *f* *'*(*c*) = 0.

The statement is **true**.

This conclusion comes from the application of the **Mean Value Theorem**. The theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists a point \( c \) in \((a, b)\) where

\[
f'(c) = \frac{f(b) - f(a)}{b - a}.
\]

In this problem:

1. The function \( f \) is given as differentiable, which implies it is also continuous.
2. We have \( f(-6) = f(6) \), meaning that the values of \( f \) at \( x = -6 \) and \( x = 6 \) are the same.

Since \( f(6) = f(-6) \), we have

\[
f'(c) = \frac{f(6) - f(-6)}{6 - (-6)} = \frac{0}{12} = 0.
\]

Therefore, there must exist a point \( c \) in the interval \( (-6, 6) \) (where \( |c| < 6 \)) such that \( f'(c) = 0 \), as required.

Would you like more details on this solution or have any questions?

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Here are 5 related questions for further exploration:

1. What are the conditions of the Mean Value Theorem and how can they be verified?
2. Can a function be continuous but not differentiable? Give examples.
3. How does the Mean Value Theorem apply to functions that are not differentiable at certain points within the interval?
4. If \( f'(c) = 0 \) at a certain point \( c \), does it necessarily mean that \( f \) has a local maximum or minimum at \( c \)?
5. How is Rolle’s Theorem a special case of the Mean Value Theorem?

**Tip:** The Mean Value Theorem is a powerful tool for proving the existence of points with specific derivative properties in an interval.

Determine whether the statement is true or false. If f is differentiable and f(−6) = f(6), then there is a number c such that |c| < 6 and f'(c) = 0.

Explore a calculus problem that uses the Mean Value Theorem to determine if a differentiable function f, with f(−6) = f(6), has a point c where |c| < 6 and f'(c) = 0. This solution demonstrates key calculus concepts and the Mean Value Theorem, ideal for students in Grades 11-12.

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