Answer is: option3

There exist \( c \) on \( (-1,5) \), such that \( f'(c) = 0 \).

Solution:

Given Information:

1. \( f \) is continuous on \([-1,5]\).
2. \( f \) is differentiable on \((-1,5)\).
3. \( f(-1) = 4 \) and \( f(5) = -2 \).

We need to determine which of the statements could be false.

(A) There exists \( c \) in \([-1,5]\) such that \( f(c) \leq f(x) \) for all \( x \) in \([-1,5]\).

1. This statement claims the existence of an absolute minimum.
2. Since \( f \) is continuous on a closed interval, the Extreme Value Theorem guarantees that \( f \) attains both a maximum and a minimum somewhere in \([-1,5]\).
3. Thus, (A) must be true and cannot be false.

(B) There exists \( c \) in \((-1,5)\) such that \( f(c) = 0 \).

1. We check whether \( f \) satisfies the Intermediate Value Theorem (IVT).
2. Since \( f(-1) = 4 \) and \( f(5) = -2 \), by IVT, there must be some \( c \) in \((-1,5)\) such that \( f(c) = 0 \) (since the function must cross the x-axis at least once).
3. Thus, (B) must be true and cannot be false.

(C) There exists \( c \) in \((-1,5)\) such that \( f'(c) = 0 \).

1. The function is continuous on \([-1,5]\) and differentiable on \((-1,5)\).
2. The Mean Value Theorem (MVT) applies because \( f(x) \) satisfies the conditions.
3. By MVT, there exists some \( c \) in \((-1,5)\) such that:

\[ f'(c) = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{-2 - 4}{6} = -1 \]

4. However, MVT does not necessarily guarantee that \( f'(c) = 0 \).
5. While \( f'(c) = 0 \) could happen, it is not guaranteed.
6. So, (C) could be false.

(D) There exists \( c \) in \((-1,5)\) such that \( f(c) = 2 \).

1. Again, we use IVT.
2. Since \( f(-1) = 4 \) and \( f(5) = -2 \), the function must take on all values between 4 and -2 at least once.
3. Since 2 is in the range \((-2,4)\), IVT guarantees some \( c \) in \((-1,5)\) where \( f(c) = 2 \).
4. Thus, (D) must be true and cannot be false.

Conclusion:

1. (A), (B), and (D) must be true.
2. (C) could be false, because we cannot guarantee \( f'(c) = 0 \) without additional information.

Final Answer: (C) could be false.