Question

Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",",  "if"  x ≤ 2),(5 - x",",  "if"  x > 2):}` at x = 2

Answer 1

f(x) is differentiable at x = 2 if Lf'(2) = Rf'(2)

∴ Lf'(2) = `lim_("h" -> 0) ("f"(2 - "h") - "f"(2))/(-"h")`

= `lim_("h" -> 0) ((1 + 2 - "h") - (1 + 2))/(-"h")`

= `lim_("h" -> 0) (3 - "h" - 3)/(-"h')`

= `(-"h")/(-"h")`

= 1

Rf'(2) = `lim_("h" -> 0) ("f"(2 + "h") - "f"(2))/"h"`

= `lim_("h" -> 0) ([5 - (2 + "h")] - (1 + 2))/"h"`

= `lim_("h" -> 0) (3 - "h" - 3)/"h"`

= `(-"h")/"h"`

= –1

So, Lf'(2) ≠ Rf'(2)

Hence, f(x) is not differentiable at x = 2.