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Question
Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",", "if" x ≤ 2),(5 - x",", "if" x > 2):}` at x = 2
Solution
f(x) is differentiable at x = 2 if Lf'(2) = Rf'(2)
∴ Lf'(2) = `lim_("h" -> 0) ("f"(2 - "h") - "f"(2))/(-"h")`
= `lim_("h" -> 0) ((1 + 2 - "h") - (1 + 2))/(-"h")`
= `lim_("h" -> 0) (3 - "h" - 3)/(-"h')`
= `(-"h")/(-"h")`
= 1
Rf'(2) = `lim_("h" -> 0) ("f"(2 + "h") - "f"(2))/"h"`
= `lim_("h" -> 0) ([5 - (2 + "h")] - (1 + 2))/"h"`
= `lim_("h" -> 0) (3 - "h" - 3)/"h"`
= `(-"h")/"h"`
= –1
So, Lf'(2) ≠ Rf'(2)
Hence, f(x) is not differentiable at x = 2.
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