Sets, Relations & Functions
Grade 11
Question:
<p>\(f(x)=x^2\sin(\pi/x^2)\) for \(x\ne 0\), \(f(0)=0\). Which is true?</p>
infinitely many solutions in [1/100,∞)
no solutions in [1/π,∞)
finite solutions in (0,1/10^{10})
more than 25 solutions in (1/π^2,1/π)
Step-by-Step Solution
<div class="solution"><p>f(x)=0 ↔ sin(\pi/x^2)=0 ↔ x=1/\sqrtn, n\inℕ. In (1/\pi^2,1/\pi): \pi^2<n<\pi^4 i.e. 10\len\le97 \to 88 solutions >25. (D) true.</p><p><strong>Answer: (D)</strong></p><div class="trap-box"><strong>Trap:</strong> Zeros come from \pi/x^2=n\pi exactly.<div class="key-concept"><strong>Key Concept:</strong> Oscillatory sin(1/x^2) -- zeros via integer counting in intervals
Correct Answer: 4