Limits, Continuity & Differentiability
Differentiable Functions
Grade 12
Question:
<p>If <i>f</i> is a differentiable function satisfying <i>f</i><sup>−1</sup><i>(n) = 0</i>, ∀<i>n ≥ 1</i>, <i>n ∈ ℤ</i>, then</p>
<p>(a) <i>f(x) = 0</i>, ∀<i>x ∈ (0, 1]</i></p>
<p>(b) <i>f'(0) = 0 = f(0)</i></p>
<p>(c) <i>f(0) = 0</i> but <i>f'(0)</i> not necessarily zero</p>
<p>(d) |<i>f(x)</i>| > 1, ∀<i>x ∈ (0, 1]</i></p>
Step-by-Step Solution
Key Concept: Differentiability combined with functional equation constraints determines both function value and derivative at critical points
<p>Given that <i>f</i><sup>−1</sup><i>(n) = 0</i> for all integers <i>n ≥ 1</i>, this means <i>f(0) = 1, 2, 3, ...</i> for multiple values, which is impossible for a function. This likely refers to <i>f(1/n) → 0</i> as <i>n → ∞</i>.</p><p>By continuity and differentiability, taking limit: <i>f(0) = 0</i> and since the derivative must also satisfy this property, <i>f'(0) = 0</i>.</p>
Correct Answer: b