Limits, Continuity & Differentiability
Methods of Differentiation
Grade 12
Question:
<p>Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a)$, $f'(b)$, $f'(c)$ are in:</p>
<p>G.P.</p>
<p>H.P.</p>
<p>A.G.P.</p>
<p>A.P.</p>
Step-by-Step Solution
Key Concept: General
<b>Properties of Quadratic Functions</b><br>Let $f(x) = Ax^2 + Bx + C$. Since $f(1) = f(-1)$:<br>$A + B + C = A - B + C \Rightarrow B = 0$<br>So $f(x) = Ax^2 + C$, and $f'(x) = 2Ax$.<br>Now $f'(a) = 2Aa$, $f'(b) = 2Ab$, $f'(c) = 2Ac$.<br>Since $a, b, c$ are in A.P., $2Aa, 2Ab, 2Ac$ are also in A.P.<br>$\therefore f'(a), f'(b), f'(c)$ are in <b>A.P.</b><br><b>Key concept:</b> $f(1)=f(-1)$ forces odd-degree terms to vanish, making $f'(x)$ proportional to $x$.<br><b>Trap:</b> Students overlook that $f(1)=f(-1)$ eliminates the linear term.
Correct Answer: D