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Question
Test whether the function f(x) `{:(= 5x - 3x^2",", "for" x ≥ 1),(= 3 - x",", "for" x < 1):}` is differentiable at x = 1
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Solution
f(x) `{:(= 5x - 3x^2",", x ≥ 1),(= 3 - x",", x < 1):}`
Rf'(1) = `lim_("h" -> 0^+) ("f"(1 + "h") - "f"(1))/"h"`
= `lim_("h" -> 0^+) (5(1 + "h") - 3(1 + "h")^2 - [5(1) - 3(1)^2])/"h"`
= `lim_("h" -> 0^+) (5 + 5"h" - 3(1 + 2"h" + "h"^2) - 2)/"h"`
= `lim_("h" -> 0^+) (-"h" - 3"h"^2)/"h"`
= `lim_("h" -> 0^+) ("h"(-1 - 3"h"))/"h"`
= `lim_("h" -> 0^+) (-1 - 3"h")` ...[∵ h → 0, ∴ h ≠ 0]
= – 1
Lf'(1) = `lim_("h" -> 0^-) ("f"(1 + "h") - "f"(1))/"h"`
= `lim_("h" -> 0^-) (3 - (1 + "h") - [5(1) - 3(1)^2])/"h"`
= `lim_("h" -> 0^-) (3 - 1 - "h" - 2)/"h"`
= `lim_("h" -> 0^-)(- "h"/"h")`
= – 1 ...[∵ h → 0, ∴ h ≠ 0]
∵ Lf'(1) = Rf'(1)
∴ f is differentiable at x = 1.
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