Explanation

(a)
Let $f(x) = ax^{2} + bx + c$
$\because\ \ f(1) = f( - 1)$
$\Rightarrow a + b + c = a - b + c\ \Rightarrow \ \ b = 0$
$\therefore\ \ \ f(x) = ax^{2} + c\ \ \ \ \Rightarrow \ \ \ \ f^{'}(x) = 2ax$
$\therefore\ \ f^{'}\left( a_{1} \right) = 2aa_{1},\ \ \ \ \ f^{'}\left( a_{2} \right) = 2aa_{2},$
$f^{'}\left( a_{3} \right) = 2aa_{3}$
Now assume
$2f^{'}\left( a_{2} \right) = f^{'}\left( a_{1} \right) + f^{'}(a_{3})$
$\Rightarrow 2.2aa_{2} =$ $2aa_{1} + 2aa_{3}$
$\Rightarrow \ 2a_{2} = a_{1} + a_{3}$
$\Rightarrow \ a_{1},a_{2},a_{3}\ are\ in\ AP.\ \ \ \ \$
$\therefore f^{'}\left( a_{1} \right),f^{'}\left( a_{2} \right),f^{'}\left( a_{3} \right)\ \ \ are\ in\ AP.$