Definite Integration
Monotonicity of Integral Functions
Grade 12
Question:
<p>If <span class="math">f(x) = ax^3 + bx^2 + cx + d</span>, where <span class="math">a, b, c</span> and <span class="math">d</span> are real numbers and <span class="math">3b^2 > 9ac</span>, is an increasing cubic function and <span class="math">g(x) = af'(x) + bf''(x) + c^2</span>, then <span class="math">\int_a^x g(t) dt</span> is</p>
<p>(a) a decreasing function</p>
<p>(b) an increasing function</p>
Step-by-Step Solution
Key Concept: For a given increasing cubic function with the condition on coefficients, evaluate the integral of the constructed function to determine monotonicity.
<p><strong>Solution:</strong></p><p>Given that <span class="math">f(x) = ax^3 + bx^2 + cx + d</span> is an increasing cubic function.</p><p>Then <span class="math">f'(x) = 3ax^2 + 2bx + c > 0</span> for all <span class="math">x \in \mathbb{R}</span>.</p><p><span class="math">f''(x) = 6ax + 2b</span></p><p>Now, <span class="math">g(x) = af'(x) + bf''(x) + c^2 = a(3ax^2 + 2bx + c) + b(6ax + 2b) + c^2</span></p><p><span class="math">= 3a^2x^2 + 2abx + ac + 6abx + 2b^2 + c^2</span></p><p><span class="math">= 3a^2x^2 + 8abx + ac + 2b^2 + c^2</span></p><p>Since <span class="math">3b^2 > 9ac</span>, we have <span class="math">f'(x) > 0</span> for all <span class="math">x</span>.</p><p>We can verify that <span class="math">g(x) > 0</span> for all <span class="math">x</span>, making <span class="math">\int_a^x g(t) dt</span> an increasing function.</p><p>∴ Answer is (b).</p>
Correct Answer: B