Let f(x) be a polynomial of second degree. If f(1) = f(-1) and a, b, care in A.P., then f'(a), f'(b), f'(c) are in

Arithmetic-Geometric Progression AP GP HP

AP

Let $f(x)=p x^2+q x+r$. Then, $f(1)=f(-1) \Rightarrow p+q+r=p-q+r \Rightarrow q=0$ ∴  $f(x)=p x^2+r$ $\Rightarrow f'(x)=2 p x \Rightarrow f'(a)=2 a p, f'(b)=2 b p$ and $f'(c)=2 c p$ Now, a, b, c are in A.P. ⇒  2ap, 2bp, 2cp are in AP. ⇒  f'(a), f'(b), f'(c) are in AP.