Question

Why is \[|x|\] not differentiable at \[x=0\]?

Options

• \[\text{LHD}=-1\ne1=\text{RHD}\]

• \[|x|\] is not continuous at \[x=0\].

• Its left-hand derivative and right-hand derivative are both \[0\].

• Its derivative exists only because \[-1=1\].

Answer 1

At \[x=0\], the left-hand derivative of \[|x|\] is \[-1\], while the right-hand derivative is \[1\]. Since these one-sided derivatives are unequal, \[|x|\] is not differentiable there.