Q81 / 150 · MHT CET 2026 (18 April Shift 2) · Mathematics
If $f : R \to R$ is an even function then
- A$f'(0) = 1$
- B$f'(x)$ is an even function
- C$f(0) = 0$
- Correct answerD$f'(x)$ is an odd function
Solution
Given that $f(x)$ is an even function, we have:
$f(-x) = f(x)$
Differentiating both sides with respect to $x$, we get:
$-f'(-x) = f'(x)$
$f'(-x) = -f'(x)$
This condition satisfies the definition of an odd function. Therefore, $f'(x)$ is an odd function.
Answer: $f'(x)$ is an odd function
$f(-x) = f(x)$
Differentiating both sides with respect to $x$, we get:
$-f'(-x) = f'(x)$
$f'(-x) = -f'(x)$
This condition satisfies the definition of an odd function. Therefore, $f'(x)$ is an odd function.
Answer: $f'(x)$ is an odd function
Hence, the correct answer is option: 'D'.