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If f(x)=|x-2| write whether f\ '\ (2) ex...

If `f(x)=|x-2|` write whether `f\ '\ (2)` exists or not.

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Given that,
`f(x)=|x-2|`
`f(x)={{:(-(x−2),if xlt2),((x−2),if xge2):}`
For differentiability at `x=2`
(LHD at x=2)=`lim_(x->2^-)(f(x)-f(2))/(x-2)`
`=lim_(h->0)(f(2-h)-f(2))/(2-h-2)`
`=lim_(h->0)(-(2-h-2))/-h`
`lim_(h->0)h/-h`
`=-1`
Now
(RHD at x=2)=`lim_(x->2^+)(f(x)-f(2))/(x-2)`
`=lim_(h->0)(f(2+h)-f(2))/(2+h-2)`
`=lim_(h->0)((2+h-2))/h`
`lim_(h->0)h/h`
`=1`
so,`LHDneRHD`
so,`f'(2) `does not exist.
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