<p>If \(f(x)\) and \(g(x)\) have no derivative at \(x = a\), then \(f(x) + g(x)\) may have a derivative at \(x = a\).</p><p><strong>State whether the statement is true or false.</strong></p>
Step-by-Step Solution
Key Concept: A sum of two non-differentiable functions can sometimes be differentiable if their non-differentiabilities cancel each other out at that point. For example, if f'(a⁺) ≠ f'(a⁻) but g'(a⁺) and g'(a⁻) are their exact negatives, the sum becomes differentiable.
<p><strong>Step 1:</strong> Recall that f(x) is differentiable at x=a if and only if f'(a⁻) = f'(a⁺) exists.</p><p><strong>Step 2:</strong> Consider a counterexample: Let f(x) = |x| and g(x) = -|x| at x = 0.</p><p><strong>Step 3:</strong> f(x) has no derivative at x=0 since f'(0⁻) = -1 and f'(0⁺) = 1.</p><p><strong>Step 4:</strong> g(x) has no derivative at x=0 since g'(0⁻) = 1 and g'(0⁺) = -1.</p><p><strong>Step 5:</strong> However, f(x) + g(x) = |x| - |x| = 0 (constant function).</p><p><strong>Step 6:</strong> The constant function 0 is differentiable everywhere with derivative 0, including at x=0.</p><p><strong>Step 7:</strong> Therefore, the sum of two non-differentiable functions CAN have a derivative.</p><p>∴ The statement is <strong>TRUE</strong> (Answer: A)</p>
Correct Answer: A