Question

Determine whether the following function is differentiable at x = 3 where,

f(x) `{:(= x^2 + 2","  ,  "for"  x ≥ 3),(= 6x - 7"," ,  "for"  x < 3):}`

Answer 1

f(x) `{:( = x^2 + 2","  ,  "for"  x ≥ 3),(= 6x - 7"," ,  "for"  x < 3):}`

Differentiability at x = 3

Lf'(3) = `lim_("h" -> 0^-) ("f"("h" + 3) - "f"(3))/"h"`

= `lim_("h" -> 0^-) (6("h" + 3) - 7 - (3^2 + 2))/"h"`

= `lim_("h" -> 0^-) (18 + "6h" - 7 - 11)/"h"`

= `lim_("h" -> 0^+) "6h"/"h"`

= `lim_("h" -> 0^+) 6      ...[∵ h → 0, ∴ h ≠ 0]`

Rf'(3) = `lim_("h" -> 0^+) ("f"("h" + 3) - "f"(3))/"h"`

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

= `lim_("h" -> 0^+) ("h"^2 + 6"h" + 9 + 2 - 11)/"h"`

= `lim_("h" -> 0^+) ("h"^2 + 6"h")/"h"`

= `lim_("h" -> 0^+) ("h" + 6)      ...[∵ h → 0, ∴ h ≠ 0]`

= 6

∵ Lf'(3) = Rf'(3)

∴ f is differentiable at x = 3.