For the function \( f(x)=\sqrt{x} \) , which of the following statements are true?
(A) \( f(x) \) is continuous for \( x\ \ge 0 \).
(B) \( f(x) \) is continuous for \( x\in R \) (Set of real numbers).
(C) \( f(x) \) is differentiable for \( x\gt 0 \).
(D) \( f(x) \) is differentiable for \( x\ \in R \) (Set of real numbers).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
\( \sqrt{x} \) is defined only for x ≥ 0, and it is continuous on its whole domain [0, ∞). Its derivative is
\( \displaystyle f'(x)=\frac{1}{2\sqrt{x}} \)
which exists for x > 0 but not at x = 0.
So f is continuous only for x ≥ 0 and differentiable only for x > 0.
Checking the statements
(A) \( f(x) \) is continuous for \( x\ \ge 0 \). — Correct. [0, ∞) is its whole domain, and \( \sqrt{x} \) is continuous there.
(B) \( f(x) \) is continuous for \( x\in R \) (Set of real numbers). — Incorrect. \( \sqrt{x} \) is not even defined for x < 0.
(C) \( f(x) \) is differentiable for \( x\gt 0 \). — Correct. \( f'(x)=\frac{1}{2\sqrt{x}} \) exists for every x > 0.
(D) \( f(x) \) is differentiable for \( x\ \in R \) (Set of real numbers). — Incorrect. f is not defined for x < 0, and \( \frac{1}{2\sqrt{x}} \) does not exist at x = 0.