# Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R - Mathematics and Statistics

Sum

Discuss whether the function f(x) = |x + 1| + |x  – 1| is differentiable ∀ x ∈ R

#### Solution

f(x) = |x + 1| + |x – 1|

= – (1 + x) + (1 – x),   x < – 1

= 1 + x + 1 – x,       –1 ≤ x < 1

= x + 1 + x – 1,         x ≥ 1

i.e., f(x) {:(= - 2x",", x < -1),(= 2",", -1 ≤ x < 1),(= 2x",", x ≥ 1):}

Differentiability at x = – 1:

Lf'(– 1) = lim_("h" -> 0^-) ("f"(- 1 + "h") - "f"(- 1))/"h"

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

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

Rf'(– 1) = lim_("h" -> 0^+) ("f"(-1 + "h") - "f"(-1))/"h"

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

∵ Lf'(– 1) ≠ Rf'(– 1)

∴ f is not differentiable at x = – 1

Differentiability at x = 1:

Lf'(1) = lim_("h" -> 0^-) ("f"(1 + "h") - "f"(1))/"h"

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

Rf'(1) = lim_("h" -> 0^+) ("f"(1 + "h") - "f"(1))/"h"

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

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

∵ Lf'(1) ≠ Rf'(1)

∴ f is not differentiable at x = 1.

∴ f is not differentiable at x = – 1 and x = 1

∴ and not differentiable  ∀ x ∈ R.

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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 9 Differentiation
Miscellaneous Exercise 9 | Q II. (5) | Page 195