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Question
Show that the function f is not differentiable at x = −3, where f(x) `{:(= x^2 + 2, "for" x < - 3),(= 2 - 3x, "for" x ≥ - 3):}`
Solution
f(x) `{:(= x^2 + 2, "for" x < - 3),(= 2 - 3x, "for" x ≥ - 3):}`
L f'(– 3) = `lim_("h" -> 0^-) ("f"(- 3 + "h") - "f"(- 3))/"h"`
= `lim_("h" -> 0^-) ([(- 3 + "h")^2 + 2] - [2 - 3(- 3)])/"h"`
= `lim_("h" -> 0^-) (9 - 6"h" + "h"^2 + 2 - 11)/"h"`
= `lim_("h" -> 0^-) ("h"^2 - 6"h")/"h"`
= `lim_("h" -> 0^-) ("h" ("h" - 6))/"h"`
= `lim_("h" -> 0^-) ("h" - 6)` ...[∵ h → 0, ∴ h ≠ 0]
= – 6
R f'(– 3) = `lim_("h" -> 0^+) ("f"(- 3 + "h") - "f"(- 3))/"h"`
= `lim_("h" -> 0^+) ([2 - 3 (- 3 + "h")] - [2 - 3 (- 3)])/"h"`
= `lim_("h" -> 0^+) ((11 - 3"h") - 11)/"h"`
= `lim_("h" -> 0^+) (-3"h")/"h"`
= `lim_("h" -> 0^+) (- 3)` ...[∵ h → 0, ∴ h ≠ 0]
= – 3
∴ L f'(– 3) ≠ R f'(– 3)
∴ f is not differentiable at x = – 3.
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