High Number Punch 05

$Let the function f (x) be continuous and always greater than zero,$
$\begin{aligned} &F(t)=\frac{\iiint_{\Omega(t)} f\left(x^{2}+y^{2}+z^{2}\right) d v}{\iiint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}\\ \end{aligned}$
$\begin{aligned} G(t)=\frac{\iint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}{\int_{-t}^{t} f\left(x^{2}\right) d x} \end{aligned}$
$其中\Omega(t)=\{(x,y,z)|x^2+y^2+z^2 \leq t^2\},D(t)=\{(x,y)|x^2+y^2 \leq t^2\}.$
$(1) Discuss the monotonicity of F (t) in the interval (0, + \ infty);$
$(2) Prove that when t> 0, F (t)> \ frac {2} {\ pi} G (t).$