Find the derivative of the function \(f(x) = |x^3 - 4x^2 + 3x - 2|\) and solve for \(x\) when \(f'(x) = 0\)

Step 1 ：Step 1: Identify the inside function of the absolute value as \(g(x) = x^3 - 4x^2 + 3x - 2\).

Step 2 ：Step 2: Find the derivative of \(g(x)\), \(g'(x) = 3x^2 - 8x + 3\).

Step 3 ：Step 3: The derivative of the absolute value function is \(f'(x) = g'(x)\) when \(g(x) > 0\) and \(f'(x) = -g'(x)\) when \(g(x) < 0\). So we need to find the values of \(x\) when \(g(x) = 0\).

Step 4 ：Step 4: Solve for \(x\) in the equation \(g(x) = 0\), which gives \(x\) approximately equal to 0.554, 1.223 and 2.723.

Step 5 ：Step 5: Check the sign of \(g'(x)\) in intervals determined by the roots of \(g(x)\). We find that \(g'(x) > 0\) when \(x < 0.554\) or \(1.223 < x < 2.723\), and \(g'(x) < 0\) when \(0.554 < x < 1.223\) or \(x > 2.723\).

Step 6 ：Step 6: Therefore, \(f'(x) = g'(x)\) when \(x < 0.554\) or \(1.223 < x < 2.723\), and \(f'(x) = -g'(x)\) when \(0.554 < x < 1.223\) or \(x > 2.723\).

Step 7 ：Step 7: Set \(f'(x) = 0\) and solve for \(x\). We find that \(x\) approximately equal to 0.333 and 2.667.