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. AP
B. GP
C. HP
D. None of these

Explanation

To solve the problem, we need to show that if \( f(1) = f(-1) \) for a polynomial function \( f(x) \) of second degree and \( a_1, a_2, a_3 \) are in arithmetic progression (AP), then \( f'(a_1), f'(a_2), f'(a_3) \) are also in AP.

### Step-by-Step Solution:

1. **Assume the form of the polynomial**:
Let \( f(x) = px^2 + qx + r \), where \( p, q, r \) are constants.

**Hint**: Start by expressing the polynomial in its general form.
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