Which of the following functions \( f(x) \) are differentiable at \( x = 0 \)?

(A) \( |x| \)
(B) \( |x - 1| \)
(C) \( |\sin x| \)
(D) \( |\cos x| \)
(E) \( x^2 \)

Choose the correct answer from the options given below:

(B), (D), and (E) only

The correct answer is Option (3) → (B), (D), and (E) only

(A) $|x|$ — ❌ Not differentiable at $x = 0$ (sharp corner)

(B) $|x - 1|$ — ✔️ Differentiable at $x = 0$, since $x = 1$ is the corner point, not $x = 0$

(C) $|\sin x|$ — ❌ Not differentiable at $x = 0$

Because: $\displaystyle f(x) = |\sin x| = \begin{cases} \sin x & \text{if } x \geq 0 \\ -\sin x & \text{if } x < 0 \end{cases}$

Left-hand derivative: $\lim_{h \to 0^-} \frac{-\sin h}{h} = -1$
Right-hand derivative: $\lim_{h \to 0^+} \frac{\sin h}{h} = 1$
Since LHD $\ne$ RHD, not differentiable at $x = 0$

(D) $|\cos x|$ — ✔️ Differentiable at $x = 0$, since $\cos x$ is differentiable at 0 and non-zero there, so no sharp corner

(E) $x^2$ — ✔️ Differentiable everywhere including $x = 0$