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. A.P.
• B. G.P.
• C. H.P.
• D. arithmetico-geometric progression

Answer 1

Answer: (A)

Let $f(x)=A{x}^2+Bx+C$
$\therefore f(1)=A+B+C$
and $f(-1)=A-B+C$
$\because f(1)=f(-1)$
$\Rightarrow A+B+C=A-B+C$
$\Rightarrow B=0$
$\therefore f(x)=A{x}^2+C$
$\Rightarrow f^{\prime }(x)=2Ax$
$\Rightarrow f^{\prime }(a)=2Aa,f^{\prime }(b)=2Ab$
and $f^{\prime }(c)=2Ac$
Also, $a,b,c$ are in A.P. .
$\therefore 2Aa,2Ab,2Ac$ are in A.P.
$\Rightarrow f^{\prime }(a),f^{\prime }(b),f^{\prime }(c)$ are in A.P.