If $f<1$, $f(0)^2 + f'(0)^2=4$, exists $x_0$ s.t. $f''(x_0) + f(x_0)=0$
$f\equiv -2$ is a counter-example. So there must be an additional restriction. I will post this as a wiki, in case anyone can solve it, given a reasonable restriction.
$f\equiv -2$ is a counter-example. So there must be an additional restriction. I will post this as a wiki, in case anyone can solve it, given a reasonable restriction.