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

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.

Tardigrade Admin · Accepted Answer

Let f(x)=p x2+q x+r f (1)= f (-1) gives p + q + r = p - q + r hence q =0 Hence f(x)=p x2+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. ⇒ (D)

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

G.P.

H.P.

A.G.P.

A.P.

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

Q. 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′(a), f′(b) and f′(c) are in