Question

Determine the intervals of monotonicity of $f\left(x\right)=\log \left|x\right|.$

• A. increasing for $x>0$
• B. increasing for $x<0$
• C. decreasing for $x>0$
• D. decreasing for $x<0$

Answer 1

Answer: (A), (D)

The correct options are
A increasing for $x>0$
D decreasing for $x<0$

$f^{\prime }\left(x\right)=\frac{1}{x},x\ne 0$ , at $x=0,$ $f\left(x\right)$ is not defined.
Hence $f\left(x\right)$ is increasing for
$x>0$ since $f^{\prime }\left(x\right)>0\forall x\in (0,\infty )$
and $f\left(x\right)$ is decreasing for $x<0$ since $f^{\prime }\left(x\right)<0\forall x\in (-\infty ,0)$