Question

The set of points where the function $f\left(x\right)=|x-2|\cos x$ is differentiable, is :

• A. $(-\infty ,\infty )$
• B. $(-\infty ,\infty )-\left\{2\right\}$
• C. $(0,\infty )$
• D. None of these

Answer 1

Answer: (B)

$(-\infty ,\infty )-\left\{2\right\}$

$f\left(x\right)=|x-2|\cos x$

mod function can be defined as,

$f\left(x\right)=(x-2)\cos x,x\ge 2$

$f\left(x\right)=(2-x)\cos x,x<2$

1. differentiability at $x=2$

LHD = $-\cos x+(2-x)(-\sin x)=-\cos 2$

RHD = $\cos x+(x-2)(-\sin x)=\cos 2$

$\therefore LHD\ne RHD$

function is not differentiable at $x=2$

2. $f\left(x\right)$ is differentiable for $x>2$

3. $f\left(x\right)$ is differentiable for $x<2$

$\therefore$ differentiable set is $(-\infty ,\infty )-\left\{2\right\}$