Limits, Continuity & Differentiability
Derivative of Even Functions
Grade 12
Question:
<p>Let <i>f</i> be an even function and \(f'(0)\) exists, then \(f'(0)\) is</p>
<p>(a) 1</p>
<p>(b) 0</p>
<p>(c) ±1</p>
<p>(d) ±2</p>
Step-by-Step Solution
Key Concept: If an even function is differentiable at 0, its derivative at 0 must be 0 because the derivative of an even function is odd, and the only value that equals its own negative is 0.
<p><strong>Step 1:</strong> Since <i>f</i> is an even function, we have $f(-x) = f(x)$ for all <i>x</i>.</p><p><strong>Step 2:</strong> Differentiating both sides with respect to <i>x</i>:<br/>$-f'(-x) = f'(x)$<br/>This means $f'(-x) = -f'(x)$, so $f'$ is an odd function.</p><p><strong>Step 3:</strong> At $x = 0$:<br/>$f'(-0) = -f'(0)$<br/>$f'(0) = -f'(0)$<br/>$2f'(0) = 0$<br/>$f'(0) = 0$</p><p>∴ Answer is (b).</p>
Correct Answer: B