Question:

Step 1: Interpret the symmetry condition. $f(3-x) = f(3+x)$ means $f$ is symmetric about $x = 3$. Step 2: Use symmetry of roots. If $x_i$ is a root, then $6 - x_i$ is also a root (reflection about $x=3$). Since there are exactly 5 distinct roots and 5 is odd, one root must be at the axis of symmetry: $x_3 = 3$. Step 3: Pair the remaining roots. $x_1$ and $x_5 = 6 - x_1$ are paired, and $x_2$ and $x_4 = 6 - x_2$ are paired. Step 4: Verify option (a). $x_1 + x_2 + x_3 + x_4 + x_5 = x_1 + x_2 + 3 + (6-x_2) + (6-x_1) = 3 + 6 + 6 = 15$. ✓ Step 5: Verify option (b). Since $x_3 = 3$ is the axis of symmetry and $f$ is symmetric about $x=3$, $x=3$ is a local extremum or inflection point of $f$. Since $f(x_3) = 0$ and $f$ is symmetric, $f'(x_3) = 0$ (the function touches zero at the axis of symmetry with zero derivative). ✓ Step 6: Verify option (c). At a simple root $x_i$ of $f$, $f$ changes sign, so $|f(x)|$ has a corner (non-differentiable point). At $x_3 = 3$, since $f'(x_3) = 0$, $|f(x)|$ is differentiable at $x_3$. Thus $|f(x)|$ is not differentiable at $x_1, x_2, x_4, x_5$ (assuming these are simple roots). ✓ Step 7: Verify option (d). Since $|f(x)|$ is not differentiable at $x_1, x_2, x_4, x_5$, it is NOT a differentiable function. Option (d) is incorrect. Step 8: Conclude. Correct options are (a), (b), (c), i.e., options 1, 2, 3.