Question

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

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

Answer 1

Answer: (D)

A.P.

Let $f\left(x\right)={px}^2+qx+r$
Differentiatin g w.r. to $x$, we get
$f^{{}^{\prime }}\left(x\right)=2px+q$
Now,
$f(-1)=f\left(1\right)$
$\therefore p-q+r=p+q+r$
$\therefore q=0$
$\therefore f^{{}^{\prime }}\left(x\right)=2px$
$f^{{}^{\prime }}\left(a\right)=2pa.f^{{}^{\prime }}\left(b\right)=2pb,f^{{}^{\prime }}\left(c\right)=2pc$
Since $a,b,c$ are in $A.P.$, therefore
$2b=a+c$
$4pb=2pa+2pc$
$2f^{{}^{\prime }}\left(b\right)=f^{{}^{\prime }}\left(a\right)+f^{{}^{\prime }}\left(c\right)$
$\therefore f^{{}^{\prime }}\left(a\right),f^{{}^{\prime }}\left(b\right),f^{{}^{\prime }}\left(c\right)$ are in A.P