The set of points where the function `f9x)=x|x|` is differentiable is `(-oo,oo)` (b) `(-oo,0)uu(0,oo)` `(0,oo)` (d) `[0,oo)`

Options

A. `(-oo,oo)`
B. `(-oo,0)cup(0,oo)`
C. `(0,oo)`
D. `[0,oo)`

Answer

A

Explanation

We have, `f(x)={{:(x^(2)",",xge0),(-x^(2)",",x lt 0):}`
Clearly, f(x) is differentiable for all xgt0 and for all xlt0. so, we check the differentiable at x=0.
Now, (RHD at x=0)
`((d)/(dx)(x)^(2))_(x=0)=(2x)_(x=0)=0`
and (LHD at x=0)
`((d)/(dx)(-x)^(2))_(x=0)=(-2x)_(x=0)=0`
`therefore` (LHD at x=0)=(RHD at x=0)
so, f(x) is differentiable for all, x i.e., the set of all points where f(x) is differentiable is `(-oo,oo)`.
Alternate solution:
It is clear from the graph f(x) is differentiable everywhere.