Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let <span class="math-inline">\(|f(x)-f(y)|\le 2|x-y|^{3/2}\)</span>, <span class="math-inline">\(f(0)=1\)</span>. Find <span class="math-inline">\(\int_0^1 f^2(x)\,dx\)</span>.</p>
0
1/2
2
<strong>1</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>|f'(x)|≤lim 2|h|^{1/2}=0 so f'=0, f=constant=1. Integral=1. <strong>Answer: (4) 1</strong></p><div class="key-concept"><strong>Key Concept:</strong> Lipschitz exponent >1 forces f to be constant</div></div>
Correct Answer: 4