Question

Test whether the function f(x) `{:(= 2x - 3",", "for"  x ≥ 2),(= x - 1",", "for"  x < 2):}` is differentiable at x = 2

Answer 1

f(x) = 2x – 3, for x ≥ 2

∴ f(2) = 2(2) – 3 = 1

Now, Rf'(2) `lim_("h" -> 0^+) ("f"(2 + "h") - "f"(2))/"h"`

= `lim_("h" -> 0) ([2(2 + "h") - 3] - 1)/"h"`  ...[∵ f(x) = 2x – 3, for x ≥ 2]

= `lim_("h" -> 0) (4 + 2"h" - 3 - 1)/"h"`

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

= `lim_("h" -> 0) 2`    ...[∵ h → 0, ∴ h ≠ 0]

= 2

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

= `lim_("h" -> 0^-) ([(2 + "h") - 1] - 1)/"h"`  ...[∵ f(x) = x – 1, for x < 2]

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

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

= `lim_("h" -> 0) 1`    ...[∵ h → 0, ∴ h ≠ 0]

= 1

∴ Rf'(2) ≠ Lf'(2)

∴ f is not differentiable at x = 2.