The set of points where the function f(x) = x / (1 + |x|) is differentiable, is:
2018
The set of points where the function f(x) = x / (1 + |x|) is differentiable, is:
Answer: B. (-∞, ∞) — Write f(x) = x/(1 + |x|) as a piecewise function by splitting on the sign of x. For x ≥ 0: |x| = x, so f(x) = x/(1 + x), and f′(x) = 1/(1 + x)². For x < 0:…
- A.
(-∞, -1) ∪ (1, ∞)
- B.
(-∞, ∞)
- C.
(0, ∞)
- D.
(-∞, 0) ∪ (0, ∞)
Attempted by 2 students.
Show answer & explanation
Correct answer: B
Write f(x) = x/(1 + |x|) as a piecewise function by splitting on the sign of x.
For x ≥ 0: |x| = x, so f(x) = x/(1 + x), and f′(x) = 1/(1 + x)².
For x < 0: |x| = −x, so f(x) = x/(1 − x), and f′(x) = 1/(1 − x)².
On each open branch the denominator (1 + x for x > 0, 1 − x for x < 0) is never zero, so f is a smooth rational function and is differentiable there.
The only join point to check is x = 0. The left-hand derivative is lim(x→0⁻) 1/(1 − x)² = 1 and the right-hand derivative is lim(x→0⁺) 1/(1 + x)² = 1. They are equal, so f is differentiable at x = 0 as well, with f′(0) = 1.
Since f is differentiable at every real number, the set of points of differentiability is (−∞, ∞), i.e. all of ℝ.