Find whether the following function is differentiable at x = 1 and x = 2 or not : `f(x)={x ,x<12-x ,1lt=xlt=2-2+3x-x^2,x >2`

Explanation

Given function,
`f(x)={{:(x ,x<12),(-x ,1lexle2),(-2+3x-x^2,x >2):}`
Left hand limit of f(x) at `x=1`
`lim_(h->0)
(1−h)=1`
Right hand limit of f(x) at `x=1`
`lim_(h->0)
(2−(1+h))=lim_(h->0)
(1−h)=1`
Since `LHL=RHL`

So, f(x) is continuous at `x=1`.

Left hand derivative of f(x) at `x=1`
`lim_(h->0)((1-h)-1)/((1-h)-1)`
`=1`,
Right hand derivative of f(x) at `x=1`
`lim_(h->0)({2-(1+h)}-(2-1))/((1+h)-1)=-1`
Since, `LHDneRHD`

So, f(x) is not differentiable at `x=1`.

Left hand limit of f(x) at `x=2`
`lim_(h->0)
...