Sets, Relations & Functions
General
Grade 11
Question:
<p><span class="math-inline">\(f(x)=x^2\sin(\pi/x^2)\)</span> for <span class="math-inline">\(x\ne 0\)</span>, <span class="math-inline">\(f(0)=0\)</span>. Which is true?</p>
infinitely many solutions in [1/100,infty)
no solutions in [1/pi,infty)
finite solutions in (0,1/10^{10})
more than 25 solutions in (1/pi^2,1/pi)
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f(x)=0 ↔ sin(π/x²)=0 ↔ x=1/√n, n∈ℕ. In (1/π²,1/π): π²<n<π⁴ i.e. 10≤n≤97 → 88 solutions >25. (D) true.</p><p><strong>Answer: (D)</strong></p><div class="trap-box"><strong>Trap:</strong> Zeros come from π/x²=nπ exactly.</div><div class="key-concept"><strong>Key Concept:</strong> Oscillatory sin(1/x²) — zeros via integer counting in intervals</div></div>
Correct Answer: 4