Limits, Continuity & Differentiability
Differentiable Functions
Grade 12
Question:
<p>Consider a differentiable function \(f: R \to R\) with \(f(0) = 0\) and \(f'(0) = 1\). Which of the following statements are true for any such function \(f\)?</p>
<p>\(f(x) > 0\) on \((0, q)\) for some positive \(q\).</p>
<p>\(f(x)\) is increasing on \((p, q)\) for some negative \(p\) and some positive \(q\).</p>
<p>There exists a differentiable function \(g: R \to R\) such that \(g''(x) = f(x)\) and</p>
<p>\(f'(x)\) is continuous.</p>
Step-by-Step Solution
Key Concept: Use the definition of derivative at x=0 as lim(h→0) f(h)/h = f'(0) = 1 to establish behavior near origin. This immediately tells us f(h)~h near h=0, constraining which limit statements must be true.
<p><strong>Step 1:</strong> Given: f(0)=0, f'(0)=1, and f is differentiable on ℝ.</p><p><strong>Step 2:</strong> From the derivative definition: f'(0) = lim(h→0) [f(h)-f(0)]/h = lim(h→0) f(h)/h = 1</p><p><strong>Step 3:</strong> This means for small h: f(h) ≈ h, which implies lim(h→0) f(h) = 0 (already given by continuity at 0).</p><p><strong>Step 4:</strong> Key consequences that MUST be true for ANY such f:</p><p>• lim(h→0) f(h)/h = 1 ✓ (by definition of f'(0))</p><p>• lim(h→0) [f(h)-h]/h = 0 (since f(h) = h + o(h) as h→0) ✓</p><p>• f is continuous at x=0 ✓ (differentiability implies continuity)</p><p>• lim(x→0) f(x)/x = 1 ✓ (equivalent to the derivative condition)</p><p><strong>Step 5:</strong> Statements claiming specific values of f'(x) for x≠0, or claiming f(x)=x globally, are NOT necessarily true—we only know local behavior near 0.</p><p>∴ Answer: A (The statements involving the derivative definition at 0 and local limiting behavior)</p>
Correct Answer: A