Question

Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function. If \( f \) is even, then \( f'(0) \) is equal to

• A. 1
• B. 2
• C. 0
• D. -1
• E. \( \frac{1}{2} \)

Hint:

Derivative of an even function is always an odd function, so value at 0 must be zero.

Answer 1

The Correct Option is C

Concept: For an even function: \[ f(-x) = f(x) \] Differentiating both sides: \[ -f'(-x) = f'(x) \Rightarrow f'(-x) = -f'(x) \] So derivative of even function is odd.

Step 1: Apply property at \( x = 0 \).
\[ f'(0) = -f'(0) \]

Step 2: Solve.
\[ 2f'(0) = 0 \Rightarrow f'(0) = 0 \] Final Answer: \[ 0 \]