If \( f(x) = x^{2}+bx+c \) and \( f(1+k) = f(1-k) \) \( \forall K \in \mathbb{R} \), for two real numbers b and c, then
Show Hint
For any quadratic function \( f(x) \), the condition \( f(a+k) = f(a-k) \) for all \(k\) immediately tells you that the parabola's axis of symmetry is at \( x=a \). This is a major shortcut.