Question:

Step 1: Recall the conditions for Rolle's Theorem. Rolle's theorem states that if a function $f$ satisfies: - $f$ is continuous on the closed interval $[a, b]$ - $f$ is differentiable on the open interval $(a, b)$ - $f(a) = f(b)$ then there exists at least one point $c \in (a, b)$ where $f'(c) = 0$. For Rolle's theorem to be applicable, all three conditions must be satisfied. Step 2: Identify the points where $f(x)$ is not differentiable. Given: $f(x) = |x^2 - 7x + 12|(x^2 - 7x + 10)(x^2 - 4x + 3)$ First, factor the expression inside the absolute value: $$x^2 - 7x + 12 = (x-3)(x-4)$$ The absolute value function $|x^2 - 7x + 12|$ is not differentiable at the points where $x^2 - 7x + 12 = 0$, which are: $$x = 3 \text{ and } x = 4$$ These are the critical points where differentiability may fail. Step 3: Check the interval $[1, 3]$. Evaluate $f$ at the endpoints: $$f(1) = |1 - 7 + 12|(1 - 7 + 10)(1 - 4 + 3) = |6|(4)(0) = 0$$ $$f(3) = |9 - 21 + 12|(9 - 21 + 10)(9 - 12 + 3) = |0|(−2)(0) = 0$$ So $f(1) = f(3) = 0$ ✓ The non-differentiable point $x = 3$ is an endpoint of the interval, not in the interior $(1, 3)$. Therefore, $f$ is differentiable on the open interval $(1, 3)$. **Conclusion:** Rolle's theorem IS applicable on $[1, 3]$. Step 4: Check the interval $[2, 4]$. Evaluate $f$ at the endpoints: $$f(2) = |4 - 14 + 12|(4 - 14 + 10)(4 - 8 + 3) = |2|(0)(−1) = 0$$ $$f(4) = |16 - 28 + 12|(16 - 28 + 10)(16 - 16 + 3) = |0|(−2)(3) = 0$$ So $f(2) = f(4) = 0$ ✓ However, $x = 3 \in (2, 4)$ is an interior point where $f$ is not differentiable. **Conclusion:** Rolle's theorem is NOT applicable on $[2, 4]$. Step 5: Check the interval $[3, 5]$. Evaluate $f$ at the endpoints: $$f(3) = |9 - 21 + 12|(9 - 21 + 10)(9 - 12 + 3) = |0|(−2)(0) = 0$$ $$f(5) = |25 - 35 + 12|(25 - 35 + 10)(25 - 20 + 3) = |2|(0)(8) = 0$$ So $f(3) = f(5) = 0$ ✓ However, $x = 4 \in (3, 5)$ is an interior point where $f$ is not differentiable. **Conclusion:** Rolle's theorem is NOT applicable on $[3, 5]$. Step 6: Determine the final answer. We found that Rolle's theorem is not applicable on both $[2, 4]$ and $[3, 5]$. Since the question asks for which range Rolle's theorem is not applicable, and we have identified multiple intervals where it fails, the answer is: $$\boxed{\text{Option 4: none of these}}$$ This is because the question likely expects a single interval, but we have shown that the theorem fails on more than one of the given options.