Limits, Continuity & Differentiability
Differentiability and Lipschitz conditions
Grade 12
Question:
<p><strong>201.</strong> Let \(f: \mathbb{R} \to \mathbb{R}\) be a function such that for all \(x, y \in \mathbb{R}\), \(|f(x) - f(y)| \leq 6|x - y|^2\). If \(f(3) = 6\), then \(f(6)\) is equal to:</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 3</p>
<p>(d) 6</p>
Step-by-Step Solution
Key Concept: A function satisfying |f(x) - f(y)| ≤ 6|x - y|² must be constant because this Lipschitz-type condition forces the derivative to be zero everywhere. Use the definition of differentiability: the constraint implies f'(x) = 0 for all x.
<p><strong>Step 1:</strong> Analyze the given constraint |f(x) - f(y)| ≤ 6|x - y|².</p><p><strong>Step 2:</strong> For differentiability, consider the difference quotient. Fix x and let y = x + h:</p><p>|f(x + h) - f(x)| ≤ 6|h|²</p><p><strong>Step 3:</strong> Divide by |h|:</p><p>|[f(x + h) - f(x)]/h| ≤ 6|h|</p><p><strong>Step 4:</strong> Taking limit as h → 0:</p><p>|f'(x)| ≤ lim(h→0) 6|h| = 0</p><p>Therefore f'(x) = 0 for all x ∈ ℝ.</p><p><strong>Step 5:</strong> A function with zero derivative everywhere is constant.</p><p>Since f(3) = 6, we have f(x) = 6 for all x ∈ ℝ.</p><p><strong>Step 6:</strong> Therefore, f(6) = 6.</p><p>∴ Answer: D</p>
Correct Answer: D