Question

Let $f ( x )$ be a polynomial function of second degree. If $f (1)= f (-1)$ and $a , b , c$ are in A.P., then $f ^{\prime}( a )$, $f^{\prime}(b)$ and $f^{\prime}(c)$ are in

• A. G.P.
• B. H.P.
• C. A.G.P.
• D. A.P.

Answer 1

Answer: (D)

Let $ f(x)=p x^2+q x+r$
$f (1)= f (-1)$ gives $p + q + r = p - q + r$
hence $q =0$
Hence $f(x)=p x^2+r$
$f^{\prime}(x)=2 p x$
Given $a, b, c$ are in A.P.
hence $2 pa , 2 pb , 2 pc$ will also be in A.P.
or $ f^{\prime}(a), f^{\prime}(b), f^{\prime}(c)$ will also be in A.P. $\Rightarrow (D)$