A rational function $f(x)$ and its first and second derivatives are given below:

$f(x)=\frac{1}{x^2-4},f^{\text{'}}(x)=\frac{-2x}{(x-2{)}^2(x+2{)}^2},f^{\text{'}\text{'}}(x)=\frac{2(3x^2+4)}{(x-2{)}^3(x+2{)}^3}$

Which of the following is FALSE about the function $f(x)?$

$f(x)$ has a local maximum at $(0,f(0)).$

$f(x)$ has no points of inflection.

$f(x)$ is concave down over the interval $(-2,2)$

$f(x)$ is increasing over the intervals $(0,2)$ and $(2,).$

$f(x)$ has horizontal asymptote $y=0$