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: Use the Lipschitz-type constraint |f(x) - f(y)| ≤ 6|x - y|² to bound f(6) - f(3), then recognize that f must be constant (differentiability forces f'(x) = 0 everywhere) by taking limits along the constraint.
<p><strong>Step 1:</strong> From the given condition |f(x) - f(y)| ≤ 6|x - y|², we can investigate the derivative.</p><p><strong>Step 2:</strong> Consider the difference quotient: |f(x+h) - f(x)|/|h| ≤ 6|h|.</p><p><strong>Step 3:</strong> Taking the limit as h → 0: |f'(x)| ≤ lim(h→0) 6|h| = 0.</p><p><strong>Step 4:</strong> This implies f'(x) = 0 for all x ∈ ℝ, so f is constant on ℝ.</p><p><strong>Step 5:</strong> Since f(3) = 6 and f is constant, 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