Explanation

(a)
Let $f(x) = Ax^{2} + Bx + C$
$\therefore\ f(1) = A + B + C$
and $f( - 1) = A - B + C$
$\because f(1) = f( - 1)$
$\Rightarrow A + B + C = A - B + C$
$\Rightarrow B = 0$
$\therefore f(x) = Ax^{2} + C$
$\Rightarrow f^{'}(x) = 2Ax$
$\Rightarrow f^{'}(a) = 2Aa,\ f^{'}(b) = 2Ab\ and\ f^{'}(c) = 2Ac$
Also, $a,b,c$ are in AP
$\therefore 2Aa,\ 2Ab,\ 2Ac$ are in AP
$\Rightarrow \ f^{'}(a),f^{'}(b),f^{'}(c)$ are in AP