Let f(x) be a polynomial function of second degree and f(1) = f(-1). If a, b, c are in AP, then f'(a), f'(b), f'(c) are in:
Suppose the 2-degree polynomial is \[ f(x) = p x^2 + q x + r \] Given that \( f(1) = f(-1) \) \[ \Rightarrow p + q + r = p - q + r \Rightarrow q = 0 \] Hence, \[ f'(x) = 2 p x \] Then, \[ f'(a), f'(b), f'(c) \text{ will be } 2 p a, 2 p b, \text{ and } 2 p c \] Since \( a, b, c \) are in Arithmetic Progression (AP), their derivatives \( 2 p a, 2 p b, 2 p c \) will also be in AP.
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