Let \(f(x) = |x - 2|\), \(x \in \mathbf{R}\). Consider the following statements:
(A) f is continuous at \(x = 2\).
(B) f is differentiable at \(x = 2\).
(C) \(f'(3) = 1\)
(D) \(f'(1) = 1\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
Removing the modulus:
\(f(x) = x - 2\) for \(x \ge 2\)
\(f(x) = 2 - x\) for \(x \lt 2\)
Both pieces give the value 0 at x = 2, so the left and right limits are both 0 = f(2).
The left and right derivatives at x = 2 are the slopes of the two pieces:
\(Lf'(2) = -1\), \(Rf'(2) = 1\)
So f is continuous at 2 but not differentiable there, f′(x) = 1 for x > 2 and f′(x) = −1 for x < 2.
Checking the statements
(A) f is continuous at \(x = 2\). — Correct. Both one-sided limits equal f(2) = 0.
(B) f is differentiable at \(x = 2\). — Incorrect. The left derivative is −1 and the right derivative is 1.
(C) \(f'(3) = 1\) — Correct. Near x = 3, f(x) = x − 2, whose derivative is 1.
(D) \(f'(1) = 1\) — Incorrect. Near x = 1, f(x) = 2 − x, so f′(1) = −1.