The problem involves applying the Mean Value Theorem (MVT) under the given condition $2 \leq f'(x) \leq 6$ for all $x$. Here's the step-by-step explanation:

Mean Value Theorem (MVT)

The Mean Value Theorem states: $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c \in (a, b)$. If $f'(x)$ is bounded between 2 and 6 for all $x$, this means: $2 \leq \frac{f(b) - f(a)}{b - a} \leq 6.$

By rearranging this inequality, we find that: $2(b - a) \leq f(b) - f(a) \leq 6(b - a).$

Analyzing the Choices

We evaluate the conditions provided in the problem:

1. Choice (a): $1 \leq f(5) - f(1) \leq 4$

   • Here, $b = 5$ and $a = 1$, so $b - a = 4$.
   • Using $2(b - a) \leq f(b) - f(a) \leq 6(b - a)$: $2(4) \leq f(5) - f(1) \leq 6(4),$ $8 \leq f(5) - f(1) \leq 24.$
   • Therefore, this choice is false because $1 \leq f(5) - f(1) \leq 4$ is not valid.

2. Choice (b): $6 \leq f(7) - f(4) \leq 18$

   • Here, $b = 7$ and $a = 4$, so $b - a = 3$.
   • Using $2(b - a) \leq f(b) - f(a) \leq 6(b - a)$: $2(3) \leq f(7) - f(4) \leq 6(3),$ $6 \leq f(7) - f(4) \leq 18.$
   • Therefore, this choice is true.

3. Choice (c): $2 \leq f(5) - f(1) \leq 8$

   • From the calculation for $f(5) - f(1)$ (see Choice (a)): $8 \leq f(5) - f(1) \leq 24.$
   • Thus, $2 \leq f(5) - f(1) \leq 8$ is false because $f(5) - f(1) \geq 8$.

4. Choice (d): $3 \leq f(7) - f(4) \leq 8$

   • From the calculation for $f(7) - f(4)$ (see Choice (b)): $6 \leq f(7) - f(4) \leq 18.$
   • Therefore, this choice is false because the lower bound is 6, not 3.

5. Choice (e): $2 \leq f(5) - f(1) \leq 12$

   • From the calculation for $f(5) - f(1)$ (see Choice (a)): $8 \leq f(5) - f(1) \leq 24.$
   • Therefore, this choice is false because $f(5) - f(1) \geq 8$.

Correct Answer:

The only correct choice is (b): $6 \leq f(7) - f(4) \leq 18$.

Would you like me to explain any step in more detail?
Here are 5 related questions to expand your understanding:

1. How does the Mean Value Theorem relate to the derivative's bounds?
2. What happens if the derivative is not bounded within a specific interval?
3. How can we generalize this problem for other intervals?
4. What are some practical applications of the Mean Value Theorem?
5. Can a function violate the Mean Value Theorem? If so, under what conditions?

Tip: Always verify the interval length $(b-a)$ before applying MVT inequalities!