Which of the following functions \(f(x)\) are differentiable at \(x = 0\)?
(A) \(|x|\)
(B) \(|x-1|\)
(C) \([x]\), where \([t]\) denotes the greatest integer \(\le t\)
(D) \(|x + 1|\)
(E) \(x^2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (D) and (E) only
A function is differentiable at a point when its left-hand and right-hand derivatives there exist and are equal. A modulus function has a corner only where the expression inside it is zero.
(A) \(|x|\) — Incorrect. \(|x| = -x\) for x < 0 and \(|x| = x\) for x > 0:
\(\text{LHD} = -1,\quad \text{RHD} = 1\)
The two differ, so there is a corner at x = 0.
(B) \(|x-1|\) — Correct. Near x = 0, x − 1 < 0, so \(|x - 1| = 1 - x\), whose derivative is −1. The corner is at x = 1, not at 0.
(C) \([x]\) — Incorrect. \([x] = -1\) for \(-1 \le x \lt 0\) and \([x] = 0\) for \(0 \le x \lt 1\). It jumps at x = 0, so it is not even continuous there, and hence not differentiable.
(D) \(|x + 1|\) — Correct. Near x = 0, x + 1 > 0, so \(|x + 1| = x + 1\), whose derivative is 1. The corner is at x = −1.
(E) \(x^2\) — Correct. It is a polynomial, with derivative 2x, which is 0 at x = 0.
So (B), (D) and (E) are differentiable at x = 0.