For the function \(f(x) = \log x\), \(x \gt 0\):
(A) \(f'(x) = \displaystyle\frac{1}{x}\)
(B) f is strictly increasing on \((0, \infty)\)
(C) f has a local maximum at \(x = 1\)
(D) \(x = 1\) is a critical point of f
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Differentiating:
\( \displaystyle f'(x) = \frac{1}{x} \)
For x > 0, \(\displaystyle\frac{1}{x} \gt 0\), and it is never zero.
So \(f'(x) = \frac{1}{x}\) is positive for every x > 0, and f has no critical point.
Checking the statements
(A) \(f'(x) = \displaystyle\frac{1}{x}\) — Correct. Standard derivative of \(\log x\).
(B) f is strictly increasing on \((0, \infty)\) — Correct. \(\frac{1}{x} \gt 0\) for all x > 0.
(C) f has a local maximum at \(x = 1\) — Incorrect. f keeps increasing through x = 1, since f’(1) = 1 > 0.
(D) \(x = 1\) is a critical point of f — Incorrect. f’(1) = 1 ≠ 0; there f(1) = 0, but that is a zero of f, not of f’.