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 $≤t$
(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) $|x|$ → Not differentiable at $x=0$ (corner point)
(B) $|x-1|$ → Differentiable at $x=0$ (no corner at $0$)
(C) $[x]$ → Not differentiable at integers, including $x=0$
(D) $|x+1|$ → Differentiable at $x=0$ (corner at $x=-1$, not $0$)
(E) $x^{2}$ → Differentiable everywhere
Hence differentiable at $x=0$: (B), (D), (E)