Question

Which of the following functions is differentiable at $x = 0$ ?

• A. $\cos(|x|) + |x|$
• B. $\cos(|x|) - |x|$
• C. $\sin(|x|) + |x|$
• D. $\sin(|x|) - |x|$

Answer 1

Answer: (D)

$\sin(|x|) - |x|$

Answer 2

Let $f(x) = \sin(|x|) - |x|$
$\Rightarrow f\left(x\right) = \begin{cases} \sin \: x - x, x > 0 \\\ \- \sin \: x + x, x > 0 \end{cases}$
$\Rightarrow f'\left(x\right) = \begin{cases} \cos\: 1 - x, x \geq 0 \\\ \- \cos\: x + 1, x > 0 \end{cases}$
$\therefore$ L.H.D = $\lim_{x\to0^{-}}\left(-\cos x+1\right) = -1+1=0$
R.H.D = $\lim_{x\to0^{+}}\left(\cos x - 1\right) = -1 - 1=0$
$\because$ L.H.D = R.H.D
$\therefore \:\:\: f(x)$ is differentiable at $x = 0$