Question

Test whether the function f(x) `{:(= 5x - 3x^2",", "for"  x ≥ 1),(= 3 - x",", "for"  x < 1):}` is differentiable at x = 1

Answer 1

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.