Question

If \(f:R\rightarrow R\) is an even function then

• A. \(f^'(0) = 1\)
• B. \(f^'(x)\) is an even function
• C. \(f(0) = 0\)
• D. \(f^'(x)\) is an odd function

Hint:

Differentiating f(-x) = f(x) shows f'(-x) = -f'(x), so f' is odd.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A function is even when \(f(-x) = f(x)\) for all \(x\). Its graph is symmetric about the Y-axis. We find the symmetry of the derivative.

Step 2: Key Formula or Approach:
Differentiate both sides of \(f(-x) = f(x)\) using the chain rule.

Step 3: Detailed Explanation:
Differentiate: \(-f'(-x) = f'(x)\), so
\[ f'(-x) = -f'(x) \]
This is the definition of an odd function, so \(f'\) is odd. Example: \(f(x) = x^2\) is even and \(f'(x) = 2x\) is odd.
Now check the other options.
(A) \(f'(0) = 1\) is not necessary. For \(f(x) = x^2\), \(f'(0) = 0\).
(B) says \(f'\) is even, but we showed it is odd.
(C) \(f(0) = 0\) is not required: \(f(x) = x^2 + 1\) is even and \(f(0) = 1\).

Final Answer:
The derivative of an even function is an odd function, option (D). \[ \boxed{f'\text{ is odd (D)}} \]