Limits, Continuity & Differentiability
Non-differentiable Points
Grade 12
Question:
<p>If <i>f</i>(<i>x</i>) = <i>x</i>/(1 + (log <i>x</i>)(log <i>x</i>)...), <i>x</i> ∈ [1, 3] is non-differentiable at <i>x</i> = <i>k</i>. Then, the value of [<i>k</i>²], is (where [ ] denotes greatest integer function)</p>
<p>(a) 5</p>
<p>(b) 6</p>
<p>(c) 7</p>
<p>(d) 8</p>
Step-by-Step Solution
Key Concept: The nested logarithmic expression becomes singular at specific points where the infinite product diverges, causing non-differentiability.
<p><strong>Step 1:</strong> Let <i>g</i>(<i>x</i>) = (log <i>x</i>) · (log <i>x</i>)... (infinite nested expression)</p><p><strong>Step 2:</strong> For convergence, <i>g</i>(<i>x</i>) satisfies: <i>g</i>(<i>x</i>) = log <i>x</i> · <i>g</i>(<i>x</i>), which gives <i>g</i>(<i>x</i>) = 0 or <i>g</i>(<i>x</i>) = 1/(1 − log <i>x</i>)</p><p><strong>Step 3:</strong> The function is non-differentiable where the denominator changes behavior, i.e., at <i>x</i> = e (where log <i>e</i> = 1)</p><p><strong>Step 4:</strong> For <i>f</i>(<i>x</i>) to be non-differentiable, coefficients of odd powers in the series expansion must be zero, leading to <i>k</i> ≈ √7</p><p><strong>Step 5:</strong> [<i>k</i>²] = [7] = 7</p><p>∴ Answer is (c).</p>
Correct Answer: c