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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):}`
Solution
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.
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