Limits, Continuity & Differentiability
Properties of Differentiable Functions
Grade 12
Question:
<p>Let \(f(x) = N(x) + O(x)\) where \(N'(a)\) and \(O'(a)\) are finite and definite. Which of the following statements is true?</p>
<p>(a) \(f(x)\) is continuous at \(x = a\)</p>
<p>(b) \(f(x)\) is differentiable at \(x = a\)</p>
<p>(c) \(f'(x)\) is continuous at \(x = a\)</p>
<p>(d) Both (a) and (b) are true</p>
Step-by-Step Solution
Key Concept: Differentiability implies continuity, and properties of differentiation are preserved under addition of differentiable functions. However, the derivative itself need not be continuous.
<p><strong>Step 1:</strong> Since $N'(a)$ and $O'(a)$ exist and are finite, both $N(x)$ and $O(x)$ are differentiable at $x = a$.</p><p><strong>Step 2:</strong> If a function is differentiable at a point, it is continuous at that point.</p><p><strong>Step 3:</strong> Therefore, $N(x)$ and $O(x)$ are both continuous at $x = a$.</p><p><strong>Step 4:</strong> The sum of continuous functions is continuous, so $f(x) = N(x) + O(x)$ is continuous at $x = a$.</p><p><strong>Step 5:</strong> The sum of differentiable functions is differentiable, so $f(x)$ is differentiable at $x = a$, with $f'(a) = N'(a) + O'(a)$.</p><p><strong>Step 6:</strong> However, the continuity of $f'(x)$ at $x = a$ is not guaranteed just because $f'(a)$ exists.</p><p>∴ Answer is (d).</p>
Correct Answer: D