<p>The number of points in (1, 3), where \(f(x) = a[x^2]\), \(a > 1\), is not differentiable, where [<i>x</i>] denotes the integral part of <i>x</i>.</p>
Step-by-Step Solution
Key Concept: A function f(x) = a[x²] is not differentiable at points where the argument x² crosses integer values, i.e., where x² = n for positive integers n. Find all such points in the open interval (1, 3).
<p><strong>Step 1:</strong> Identify where f(x) = a[x²] is not differentiable. The floor function [u] has jump discontinuities (hence non-differentiability) when u crosses integer values.</p><p><strong>Step 2:</strong> Find where x² equals a positive integer in the interval (1, 3).<br>For x ∈ (1, 3): x² ∈ (1, 9)<br>Integer values of x² in this range: 2, 3, 4, 5, 6, 7, 8</p><p><strong>Step 3:</strong> Solve for x values:<br>• x² = 2 ⟹ x = √2 ≈ 1.41 ∈ (1, 3) ✓<br>• x² = 3 ⟹ x = √3 ≈ 1.73 ∈ (1, 3) ✓<br>• x² = 4 ⟹ x = 2 ∈ (1, 3) ✓<br>• x² = 5 ⟹ x = √5 ≈ 2.24 ∈ (1, 3) ✓<br>• x² = 6 ⟹ x = √6 ≈ 2.45 ∈ (1, 3) ✓<br>• x² = 7 ⟹ x = √7 ≈ 2.65 ∈ (1, 3) ✓<br>• x² = 8 ⟹ x = 2√2 ≈ 2.83 ∈ (1, 3) ✓<br>• x² = 9 ⟹ x = 3 (boundary, excluded)</p><p><strong>Step 4:</strong> Count valid points. All seven points √2, √3, 2, √5, √6, √7, 2√2 lie strictly in (1, 3).</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7