<p>If \(f(x)\) and \(g(x)\) are not differentiable finitely at a point then will \(f(x) \cdot g(x)\) will also be nondifferentiable finitely at that point?</p>
<p>(a) Yes, always</p>
<p>(b) No, not necessarily</p>
<p>(c) Only if both are continuous</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: A product of two non-differentiable functions can be differentiable if one function has a zero at that point with sufficient multiplicative order. The non-differentiability of individual functions does not guarantee non-differentiability of their product.
<p><strong>Step 1:</strong> Consider the definition—if f(x) and g(x) are non-differentiable at a point a, it means at least one of them has a sharp corner, cusp, or discontinuity there.</p><p><strong>Step 2:</strong> Examine the counterexample: Let f(x) = |x| and g(x) = |x|. Both are non-differentiable at x = 0.</p><p><strong>Step 3:</strong> Compute their product: f(x)·g(x) = |x|·|x| = x². The function x² is differentiable everywhere, including at x = 0, with derivative 2x.</p><p><strong>Step 4:</strong> The product eliminates the non-differentiability because the zeros combine multiplicatively. When f(0) = 0 and g(0) = 0, the product's behavior near the origin can become smooth.</p><p><strong>Step 5:</strong> Therefore, the answer is <strong>NO</strong>—the product f(x)·g(x) need not be non-differentiable at that point.</p><p>∴ Answer: B (No, the product can be differentiable)</p>
Correct Answer: B