Definite Integration
Orthogonal Functions
Grade 12
Question:
<p>If \(f : [0, \pi] \to \mathbb{R}\) is continuous and \(\int_0^\pi f(x) \sin x \, dx = \int_0^\pi f(x) \cos x \, dx = 0\), then</p>
<p>(A) \(f\) has at least two zeros in \([0, \pi]\)</p>
<p>(B) \(f\) has exactly one zero in \([0, \pi]\)</p>
<p>(C) \(f\) is identically zero</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use orthogonality conditions on a continuous function to establish the existence of zeros.
<p>The two integral conditions imply that $f$ is orthogonal to both $\sin x$ and $\cos x$. By properties of continuous functions and orthogonality, $f$ must have at least two zeros.</p>
Correct Answer: A