Question

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = |x - 1|`

Answer 1

`f(x) = {{:(x - 1,  "if"  x > 1),(-(x - 1),  "if"  x < 1):}`

First we find left limit of `f(x)` at x = 1

When `x -> 1-` we have `f(x) = - (x - 1)`

`f"'"(1^-) =  lim_(x -> 1^-) (f(x) - f(1))/(x - 1)`

`f"'"(1^-) =  lim_(x -> 1) (-(x - 1) - (0))/(x - 1)`

= `lim_(x -> 1) (-(x - 1))/(x - 1)` = – 1  .......(1)

`f"'"(1^+) =  lim_(x -> 1^+) (f(x) - f(1))/(x - 1)`

= `lim_(x -> 1^+) ((x - 1) - 0)/(x - 1)`

`f"'"(1^+) =  lim_(x -> 1^+) (x - 1)/(x - 1)` = 1  ......(2)

From equation (1) and (2) we have

`lim_(x -> 1^-) f(x)  ≠  lim_(x -> 1^+) f(x)`

∴ `f"'"(x)` does not exist at x = 1