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