Test whether the function f(x) =x2+1,for x≥2=2x+1,for x<2 is differentiable at x = 2 - Mathematics and Statistics

Sum

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

Solution

f(x) = x2 + 1, for x ≥ 2

∴ f(2) = 22 + 1 = 5

Now, Lf'(2) = lim_("h" -> 0^-) ("f"(2 + "h") - "f"(2))/"h"

= lim_("h" -> 0) ([2(2 + "h") + 1] - 5)/"h"  ...[∵ f(x) = 2x + 1, for x < 2]

= lim_("h" -> 0) (4 + 2"h" + 1 - 5)/"h"

= lim_("h" -> 0) (2"h")/"h"

= lim_("h" -> 0) 2   ...[∵ h → 0, ∴ h ≠ 0]

= 2

Rf'(2) = lim_("h" -> 0^+) ("f"(2 + "h") - "f"(2))/"h"

= lim_("h" -> 0) ([(2 + "h")^2 + 1] - 5)/"h"  ...[∵ f(x) = x2 + 1, for x ≥ 2]

= lim_("h" -> 0) (4 + 4"h" + "h"^2 + 1 - 5)/"h"

= lim_("h" -> 0) (4"h" + "h"^2)/"h"

= lim_("h" -> 0) ("h"(4 + "h"))/"h"

= lim_("h" -> 0) (4 + "h")  ...[∵ h → 0, ∴ h ≠ 0]

= 4 + 0

= 4

∴ Lf'(2) ≠ Rf'(2)

∴ f is not differentiable at x = 2.

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Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 9 Differentiation
Miscellaneous Exercise 9 | Q II. (7) | Page 195