Question

Let \( f(x) \) be a second degree polynomial. If \( f(1)=f(-1) \) and \( p,q,r \) are in A.P., then \( f'(p), f'(q), f'(r) \) are:

• A. in A.P.
• B. in G.P.
• C. in H.P.
• D. neither in A.P. nor G.P. nor H.P.

Hint:

If derivative is linear: \begin{itemize} \item Linear functions preserve arithmetic progression. \item A.P. inputs ⇒ A.P. outputs. \end{itemize}

Answer 1

The Correct Option is A

Concept: Let: \[ f(x) = ax^2 + bx + c \] Given: \[ f(1) = f(-1) \] This condition restricts the polynomial. Step 1: {\color{red}Use given condition.} \[ a + b + c = a - b + c \] \[ b = 0 \] So polynomial becomes: \[ f(x) = ax^2 + c \] Step 2: {\color{red}Find derivative.} \[ f'(x) = 2ax \] This is a linear function in \( x \). Step 3: {\color{red}Use A.P. property.} If \( p,q,r \) are in A.P., then: \[ q = \frac{p+r}{2} \] Now: \[ f'(p) = 2ap, \quad f'(q)=2aq, \quad f'(r)=2ar \] Since multiplying an A.P. by constant preserves A.P., \( f'(p), f'(q), f'(r) \) are also in A.P.