The function f(x) = (x /1+|x|) is

Question

The function f(x) = (x /1+|x|) is (A) strictly increasing (B) strictly decreasing (C) neither increasing nor decreasing (D) not differentiable at x = 0.

Tardigrade Admin · Accepted Answer

f'(x) = ((1+|x|)⋅1-x(±1)/[1+|x|]2) = (1+|x|-|x|/(1+|x|)2) = (1/(1+|x|)2) > 0 ∀ x ∈ R ∴ f is strictly increasing.

Q. The function $f (x) = \frac{x}{1 + ∣ x ∣}$ is

strictly increasing

strictly decreasing

neither increasing nor decreasing

not differentiable at x = 0.

$f^{'} (x) = \frac{( 1 + ∣ x ∣ ) \cdot 1 - x ( \pm 1 )}{[ 1 + ∣ x ∣ ] ^{2}}$
$= \frac{1 + ∣ x ∣ - ∣ x ∣}{( 1 + ∣ x ∣ ) ^{2}}$
$= \frac{1}{( 1 + ∣ x ∣ ) ^{2}} > 0 \forall x \in R$
$∴ f$ is strictly increasing.

Q. The function f(x)=1+∣x∣x​ is