Quadratic Equations
Properties of Quadratic Functions
Grade 11
Question:
<p>Let <span class="math">f(x)</span> be a polynomial function of second degree. If <span class="math">f(1) = f(-1)</span> and <span class="math">a, b, c</span> are in AP, then <span class="math">f'(a), f'(b)</span> and <span class="math">f'(c)</span> are in</p>
<p>(a) AP</p>
<p>(b) GP</p>
<p>(c) HP</p>
<p>(d) Arithmetic-Geometric progression</p>
Step-by-Step Solution
Key Concept: The condition f(1) = f(-1) for a quadratic forces the linear coefficient to be zero, making f'(x) = 2Ax a linear function that preserves AP progression.
<p>Let <span class="math">f(x) = Ax^2 + Bx + C</span></p><p><span class="math">f(1) = A + B + C</span> and <span class="math">f(-1) = A - B + C</span></p><p>Given <span class="math">f(1) = f(-1)</span>:</p><p><span class="math">A + B + C = A - B + C</span></p><p><span class="math">\Rightarrow 2B = 0</span></p><p><span class="math">\Rightarrow B = 0</span></p><p>Therefore, <span class="math">f(x) = Ax^2 + C</span></p><p><span class="math">f'(x) = 2Ax</span></p><p><span class="math">f'(a) = 2Aa, \quad f'(b) = 2Ab, \quad f'(c) = 2Ac</span></p><p>Since <span class="math">a, b, c</span> are in AP, we have <span class="math">2Aa, 2Ab, 2Ac</span> are also in AP.</p><p>Hence, <span class="math">f'(a), f'(b), f'(c)</span> are in AP.</p><p>∴ Answer is (a).</p>
Correct Answer: A