Question

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

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

Answer 1

Answer: (A)

A. P.

Let $f\left(x\right)=p{x}^2+qx+r$
$f\left(1\right)=f(-1)$= p+ q +r = p-q +r
$\therefore$ 2q
2q = 0 or q=0
$f\left(x\right)=p{x}^2+r\therefore f^{\prime }\left(x\right)=2px$
$f^{\prime }\left(a\right)=2pa,f^{\prime }\left(b\right)=2pb,f^{\prime }\left(c\right)=2pc$
They are in A.P. as a, b, c are in A.P.