Question

Show that the following functions are not differentiable at the indicated value of x.

`f(x) = {{:(-x + 2, x ≤ 2),(2x - 4, x > 2):}` , x = 2

Answer 1

First we find the left limit of f(x) at x = 2

When x = 2, then x ≤ 2

∴ f(x) = – x + 2

f(2) = – 2 + 2 = 0

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

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

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

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

Next we find the right limit of f(x) at x = 2

When x = 2⁺, then x > 2

∴ f(x) = 2x – 4

f(2) = 2 × 2 – 4 = 4 – 4 = 0

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

= `lim_(x -> 2^+) (2x - 4 - 0)/(x - 2)`

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

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

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

f’(2^–) ≠ f'(2⁺)

∴ f(x) is not differentiable at x = 2