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

(1) f(x) = cos |x| + |x| = $\left\{ \begin{matrix} \cos x - x, & x < 0 \\ \cos x + x, & x \geq 0 \end{matrix} \right.\$

f’(x) = $\left{ \begin{matrix}

• \sin x - 1, & x < 0 \
• \sin x + 1, & x \geq 0 \end{matrix} \right.\ \begin{matrix} Atx = 0 \ LD = - 1 \ RD = 1 \end{matrix}$

∴ Not differentiable

(2) f(x) cos |x| − |x| = $\left\{ \begin{matrix} \cos x + x, & x < 0 \\ \cos x - x, & x \geq 0 \end{matrix} \right.\$

Not differentiable at x = 0

(3) f(x) sin |x| + |x| = $\left{ \begin{matrix}

• \sin x - x, & x < 0 \

• \sin x + x, & x \geq 0 \end{matrix} \right.\ $

Not differentiable at x = 0

(4) f(x) = sin|x| −|x| = $\left{ \begin{matrix}

• \sin x + x, & x < 0 \ \sin x - x, & x \geq 0 \end{matrix} \right.\ $

f’(x) = $\left{ \begin{matrix}

• \cos x + 1, & x < 0 \

• \cos x - 1, & x \geq 0 \end{matrix} \right.\ \begin{matrix} Atx = 0 \ LD = 0 \ RD = 0 \end{matrix}$

∴ f is differentiable at x = 0.

∴ D is correct alternative.