let `f(x)` be a polynomial function of second degree. If `f(1)=f(-1)and a_(1),a_(2),a_(3)` are in AP, then show that `f'(a_(1)),f'(a_(2)),f'(a_(3))` are in AP.

Options

A. Arthmetic-Geometric Progression
B. AP
C. GP
D. HP

Answer

B

Explanation

Let `f(x)=px^(2)+qx+r.` Then,
`f(1)=f(-1)impliesp+q+r=p-q+rimpliesq=0`
`:." "f(x)=px^(2)+r`
`implies" "f'(x)=2pximpliesf'(a)=1ap,f'(b)=2bp" and "f'(c)=2cp`
Now,
a, b, c are in A.P.
`implies" "2ap, 2bp,2cp" are in AP."`
`implies" "f'(a),f'(b),f'(c)` are in AP.