Let's analyze each statement carefully using the given function values and relevant theorems.

Given Table:

Checking the Statements:

1. Intermediate Value Theorem (IVT) - Statement A
   IVT states that if $f(x)$ is continuous on $[a, b]$ and $f(a) \leq k \leq f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = k$.

   • $f(2) = 1$, $f(3) = 14$, $f(4) = 20$, $f(5) = 31$.
   • Since 10 is between $f(2) = 1$ and $f(3) = 14$, IVT guarantees some $c \in (2,3)$ where $f(c) = 10$.
   • Thus, A is true.

2. Mean Value Theorem (MVT) - Statement B
   MVT states that if $f(x)$ is continuous and differentiable on $[a, b]$, then there exists $c$ in $(a, b)$ such that:
   $f'(c) = \frac{f(b) - f(a)}{b - a}$

   • For $[2,5]$,
     $f'(c) = \frac{f(5) - f(2)}{5 - 2} = \frac{31 - 1}{3} = \frac{30}{3} = 10.$
   • This satisfies the MVT.
   • Thus, B is true.

3. Extreme Value Theorem (EVT) - Statement C
   EVT states that a continuous function on a closed interval $[a, b]$ attains a maximum and minimum value.

   • Statement C claims that there is some $c$ such that $f(c) \leq f(x)$ for all $x$, meaning $f(c)$ is a global minimum.
   • Clearly, the minimum value is $f(2) = 1$, and EVT guarantees that it occurs at some $c = 2$.
   • Thus, C is true.

4. Extreme Value Theorem (EVT) - Statement D

   • EVT also guarantees the function attains a maximum, but statement D is incomplete and unclear, as it does not specify whether $f(c)$ is a maximum or minimum.
   • If the statement meant to say $f(c)$ is an arbitrary extreme value, then it would be true.
   • However, the wording makes it ambiguous, so it could be false.

Final Answer:

The false statement is D, due to its vague formulation.

Would you like a more detailed explanation of any part?

Here are 5 related questions to explore further:

1. What are the formal conditions for the Intermediate Value Theorem to apply?
2. How does the Mean Value Theorem differ from Rolle’s Theorem?
3. Why does the Extreme Value Theorem require a closed interval?
4. Can a function satisfy the Mean Value Theorem without being differentiable everywhere?
5. What happens if a function is discontinuous on an interval?

Tip: The Mean Value Theorem always guarantees a certain derivative value, but it does not tell us exactly where it occurs!