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

$f(x)=\frac{4}{x^2-25},f^{\text{'}}(x)=\frac{-8x}{(x-5{)}^2(x+5{)}^2},f^{\text{'}\text{'}}(x)=\frac{8(3x^2+25)}{(x-5{)}^3(x+5{)}^3}$

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

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

$f(x)$ is increasing over the intervals $(0,5)$ and $(5,).$

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

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

$f(x)$ is concave down over the interval $(-5,5)$