Question:

<b>Properties of Polynomials + A.P.</b><br> Since $f(x)$ is degree 2 and $f(1)=f(-1)$, the polynomial is symmetric about $x=0$.<br> Write $f(x) = px^2 + q$ (no linear term, since $f(1)=f(-1)\Rightarrow$ odd coefficients vanish).<br> Then $f'(x) = 2px$, which is linear.<br> Since $a, b, c$ are in A.P. (i.e., $b-a = c-b$):<br> $f'(b) - f'(a) = 2pb - 2pa = 2p(b-a)$<br> $f'(c) - f'(b) = 2pc - 2pb = 2p(c-b) = 2p(b-a)$<br> So $f'(a), f'(b), f'(c)$ are also in A.P.<br> <b>Key concept:</b> Derivative of a symmetric even polynomial is an odd linear function, preserving A.P. structure.<br> <b>Trap:</b> Assuming general $f(x)=px^2+qx+r$ — the condition $f(1)=f(-1)$ forces $q=0$.