Question:

<p><strong>Step 1: Analyze g(x) = 6x² - 18x + 8</strong></p><p>First, find where g(x) = 0: 6x² - 18x + 8 = 0 → 3x² - 9x + 4 = 0</p><p>Using the quadratic formula: x = (9 ± √(81-48))/6 = (9 ± √33)/6</p><p>g(x) has vertex at x = 18/12 = 3/2, where g(3/2) = 6(9/4) - 18(3/2) + 8 = 13.5 - 27 + 8 = -5.5</p><p>So g(x) crosses zero at two points and has minimum value -5.5.</p><p><strong>Step 2: Analyze f₁(x) = |g(x)|</strong></p><p>f₁(x) = |g(x)| has non-differentiability at the two zeros of g(x) where g(x) = 0. So f₁(x) has 2 points of non-differentiability (sharp corners where g changes sign).</p><p><strong>Step 3: Analyze f₂(x) = |f₁(x) - P₁| with P₁ = 7</strong></p><p>f₂(x) = |f₁(x) - 7| is non-differentiable where:</p><p>• f₁(x) has non-differentiability (2 points from Step 2)</p><p>• f₁(x) = 7 (additional points where the absolute value creates new corners)</p><p>Since f₁(x) ranges from 0 to ∞, the equation f₁(x) = 7 has multiple solutions. Given the parabolic nature of g(x), f₁(x) = 7 intersects the curve at 4 points.</p><p>Total non-differentiability in f₂: 2 + 4 = 6 points.</p><p><strong>Step 4: Analyze f₃(x) = |f₂(x) - P₂|</strong></p><p>f₃(x) is non-differentiable where:</p><p>• f₂(x) has non-differentiability (6 points from Step 3)</p><p>• f₂(x) = P₂ (new corners created)</p><p>For exactly 10 points of non-differentiability in f₃, we need: 6 + (number of solutions to f₂(x) = P₂) = 10</p><p>This means f₂(x) = P₂ must have exactly 4 solutions.</p><p><strong>Step 5: Determine the range of P₂</strong></p><p>The function f₂(x) oscillates with minimum value 0 (when f₁(x) = P₁ = 7) and maximum determined by the range of f₁(x).</p><p>For f₂(x) = P₂ to have exactly 4 solutions, P₂ must be in the intermediate range where the curve crosses a horizontal line 4 times.</p><p>By analyzing the behavior of f₂(x) through the nested structure, this occurs when P₂ ∈ (1, 8).</p><p><strong>∴ Answer: D</strong></p>