Question

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

f(x) = |x| + |x – 1| at x = 0, 1

Answer 1

To find the limit at x = 0

First we find the left limit of f(x) at x = 0

When x = 0^– |x| = – x and

|x – 1| = – (x – 1)

∴ When x = 0 we have

f(x) = – x – (x – 1)

f(x) = – x – x + 1 = – 2x + 1

f(0) = 2 × 0 + 1 = 1

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

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

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

f'(0^– = – 2 ……… (1)

∴ When x = 0⁺ we have

|x| = x and |x – 1| = – (x – 1)

∴ f(x) = x – (x – 1)

f(x) = x – x + 1

f(x) = 1

f(0) = 1

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

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

`f"'"(0^+)` = 0   .......(2)

From equations (1) and (2) , we get

f'(0^–) ≠ f’(0⁺)

∴ f(x) is not differentiable at x = 0.

To find the limit at x = 1

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

When x = 1, |x| = x and

|x – 1| = – (x – 1)

∴ f(x) = x – (x – 1)

f(x) = x – x + 1 = 1

f(x) = 1

f(1) = 1

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

= `limm_(x -> 1^-) (1 - 1)/(x - 1)`

`f"'"(1^-) =  lim_(x -> 0^-) 0/(x - 1)` = 0  ........(3)

When x = 1⁺ , |x| = x and

|x – 1| = x – 1

When x = 1, |x| = x and

|x – 1| = x – 1

∴ f(x) = x + x – 1 = 2x – 1

f(1) = 2 × 1 – 1 = 2 – 1 = 1

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

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

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

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

`f"'"(1^-) =  lim_(x -> 1^-) (2)` = 2   .........(4)

From equations (3) and (4), we get

f’(1^–) ≠ f'(1⁺)

∴ f(x) is not differentiable at x = 1