Question

Determine whether the following function is differentiable at the indicated values.

f(x) = |x² – 1| at x = 1

Answer 1

`f(x) = {{:(-(x^2 - 1),  "if"  x < 1),(x^2 - 1,  "if"  x> 1):}`

To find the left limit of f(x) at x = 1

When x → `1^-`

f(x) = `-(x^2 - 1)`

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

= `lim_(x -> 1^-) (-(x^2 - 1) - [-(1^2 - 1)])/(x - 1)`

= `lim_(x - 1^-) (-(x^2 - 1) - 0)/(x - 1)`

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

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

`f"'"(1^-) = -(1 + 1)` = – 2  ........(1)

To find the right limit of f(x) at x = 1

When x → `1^+`

f(x) = `(x^2 - 1)`

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

= `lim_(x -> 1^+) ((x^2 - 1) - (1^2 - 1))/(x - 1)`

= `lim_(x - 1^+) ((x^2 - 1) - 0)/(x - 1)`

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

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

`f"'"(1^+) = (1 + 1)` = 2  ........(2)

From equatios (1) and (2) we have

`f"'"(1^+)  ≠  f"'"(1^+)`

∴ f(x) is not differentiable at x = 1.