Limits, Continuity & Differentiability
Differentiability using inequality condition
Grade None
Question:
<p>If \(|f(x) - f(y)| \leq 2|x - y|^{3/2}\), then which of the following is true about \(f'(x)\)?</p>
<p>\(f'(x)\) exists and equals 2</p>
<p>\(f'(x) = 0\) for all \(x\)</p>
<p>\(f'(x)\) does not exist</p>
<p>\(f(x)\) is not continuous</p>
Step-by-Step Solution
Key Concept: Use the definition of derivative with the given Lipschitz-type condition: |f(x+h) - f(x)|/|h| ≤ 2|h|^(1/2) → 0 as h → 0, forcing f'(x) = 0 everywhere.
<p><strong>Step 1:</strong> Apply the definition of derivative at any point x:</p><p>f'(x) = lim(h→0) [f(x+h) - f(x)]/h</p><p><strong>Step 2:</strong> Use the given condition with y = x+h:</p><p>|f(x+h) - f(x)| ≤ 2|h|^(3/2)</p><p><strong>Step 3:</strong> Bound the difference quotient:</p><p>|[f(x+h) - f(x)]/h| ≤ 2|h|^(3/2)/|h| = 2|h|^(1/2)</p><p><strong>Step 4:</strong> Take the limit as h → 0:</p><p>|f'(x)| ≤ lim(h→0) 2|h|^(1/2) = 0</p><p><strong>Step 5:</strong> Conclude:</p><p>f'(x) = 0 for all x in the domain of f</p><p>∴ <strong>f is constant (or equivalently, f'(x) = 0 everywhere)</strong></p>
Correct Answer: B