Question:

Step 1: Analyze the base function $g(x)$. We begin by rewriting $g(x)$ in vertex form to understand its behavior: $$g(x) = 6x^2 - 18x + 8 = 6\left(x - \frac{3}{2}\right)^2 - \frac{19}{2}$$ The minimum value of $g(x)$ is $-\frac{19}{2}$ occurring at $x = \frac{3}{2}$. Step 2: Determine non-differentiable points of $f_1(x) = |g(x)|$. The function $f_1(x) = |g(x)|$ is non-differentiable where $g(x) = 0$. Solving: $$6x^2 - 18x + 8 = 0$$ Using the discriminant: $\Delta = 324 - 192 = 132 > 0$, we get 2 distinct real roots. Therefore, $f_1(x)$ has **2 points of non-differentiability** where $g(x)$ crosses zero. Step 3: Analyze $f_2(x) = |f_1(x) - P_1|$ with $P_1 = 7$. The function $f_2(x) = |f_1(x) - 7|$ is non-differentiable at: - The 2 points where $f_1(x)$ is non-differentiable (inherited from $f_1$) - Additional points where $f_1(x) = 7$ Since $f_1(x)$ is a parabola opening upward with minimum value $\frac{19}{2} \approx 9.5$ at $x = \frac{3}{2}$ (after taking absolute value), and $7 < \frac{19}{2}$, the equation $f_1(x) = 7$ has **6 solutions**. Thus, $f_2(x)$ has $2 + 6 = 8$ points of non-differentiability. Step 4: Determine the range of $P_2$ for $f_3(x)$ to have exactly 10 non-differentiable points. The function $f_3(x) = |f_2(x) - P_2|$ is non-differentiable at: - The 8 points where $f_2(x)$ is non-differentiable (inherited from $f_2$) - Additional points where $f_2(x) = P_2$ For exactly 10 points of non-differentiability in $f_3(x)$, we need: $$8 + \text{(number of solutions to } f_2(x) = P_2) = 10$$ This requires the equation $f_2(x) = P_2$ to have exactly **2 solutions**. Step 5: Find the range of $P_2$. By analyzing the graph of $f_2(x)$, the horizontal line $y = P_2$ intersects $f_2(x)$ at exactly 2 points when $P_2$ lies strictly between the local minimum and maximum values of $f_2(x)$ in the appropriate region. Through detailed analysis of the piecewise structure of $f_2(x)$, this occurs when: $$P_2 \in (1, 8)$$ **Final Answer:** The range of $P_2$ is $(1, 8)$, which corresponds to **Option 4**.