Question

{"given":"Function $f(x)$ satisfies the symmetry condition $f(7 - x) = f(7 + x)$ for all $x \\in \\mathbb{R}$. The function has exactly 5 distinct real roots, and we need to find the ratio $\\frac{S}{7}$ where $S$ is the sum of all roots.","key_observation":"The condition $f(7 - x) = f(7 + x)$ means that $f(x)$ is symmetric about the line $x = 7$. For any function symmetric about $x = 7$, if $a$ is a root, then $14 - a$ must also be a root (except when $a = 7$). Since we have 5 distinct roots and symmetry about $x = 7$, one root must be exactly at $x = 7$, and the remaining 4 roots form 2 symmetric pairs.","option_analysis":[{"label":"(A)","text":"1","verdict":"incorrect","explanation":"If $\\frac{S}{7} = 1$, then $S = 7$. This would mean the sum of 5 distinct roots is only 7, which contradicts our analysis showing the sum must be 35."},{"label":"(B)","text":"3","verdict":"incorrect","explanation":"If $\\frac{S}{7} = 3$, then $S = 21$. This doesn't match our calculation. With one root at 7 and two symmetric pairs summing to 28, the total sum is 35."},{"label":"(C)","text":"5","verdict":"correct","explanation":"Let the roots be $x_1, x_2, 7, x_4, x_5$ where $x_1 + x_5 = 14$ and $x_2 + x_4 = 14$ due to symmetry. Then $S = x_1 + x_2 + 7 + x_4 + x_5 = 14 + 14 + 7 = 35$, so $\\frac{S}{7} = 5$."},{"label":"(D)","text":"7","verdict":"incorrect","explanation":"If $\\frac{S}{7} = 7$, then $S = 49$. This is too large given that we have one root at 7 and symmetric pairs that contribute 28 to the sum."}],"answer":"(C)","formula_steps":[]}