Definite Integration
Integral Inequalities
Grade 12
Question:
<p>Given \(\int_a^b f(x) \, dx = \int_b^a f(x) \, dx\) and \(f'(x) \geq 0\) at any \(x \in (a,b)\), with \(f(x)\) being continuous and differentiable in \((a,b)\). If \(g(x) = \int_0^x f(t) \, dt\), then \(\int_a^b f(x)g(x) \, dx \geq 0\) implies</p>
<p>(A) \(g(x) = 0\) has at most one root in \((a, b)\)</p>
<p>(B) \(g(x) = 0\) has at least one root in \((a, b)\)</p>
<p>(C) \(g(x) = 0\) has exactly one root in \((a, b)\)</p>
<p>(D) \(g(x) = 0\) for all \(x \in (a,b)\)</p>
Step-by-Step Solution
Key Concept: Use monotonicity of $f$ and properties of the integral to establish the existence of a root for $g$.
<p>Since $f'(x) \geq 0$, $f$ is increasing. The condition $\int_a^b f(x)g(x) \, dx \geq 0$ combined with the properties of $g$ guarantees that $g$ must have at least one zero in the interval.</p>
Correct Answer: B