For every twice differentiable function $f:\mathbb{R}\to[-2,2]$ with $(f(0))^{2}+(f'(0))^{2}=85$, which of the following statement(s) is (are) TRUE?

$f: \mathbb{R} \to [-2,2] \text{ twice differentiable}$ $(f(0))^2 + (f'(0))^2 = 85$ $\text{Which of (1) - (4) is/are TRUE?}$ $f: \mathbb{R} \to [-2,2],\quad (f(0))^2 + (f'(0))^2 = 85$ $|f(0)| \le 2 \Rightarrow (f'(0))^2 = 85 - (f(0))^2 \ge 81$ $\Rightarrow |f'(0)| \ge 9 \ne 0$ $(1)\ \text{local monotonicity}$ $(2)\ \text{MVT on } [-4,0]$ $(3)\ \text{limit at } \infty;\quad (4)\ h = f^2 + f'^2$ $f'(0) \ne 0,\ f' \text{ continuous} \Rightarrow f' \text{ keeps sign near } 0$ $\Rightarrow f \text{ one-one on some } (r,s) \Rightarrow (1) \text{ TRUE}$ $\text{MVT on } [-4,0]:\ f(0)-f(-4) = 4 f'(x_0)$ $|f(0)-f(-4)| \le 4 \Rightarrow |f'(x_0)| \le 1 \Rightarrow (2) \text{ TRUE}$ $\lim_{x\to\infty} f(x) = 1 \text{ is not forced} \Rightarrow (3) \text{ FALSE}$ $h(x) = f(x)^2 + f'(x)^2,\quad h(0) = 85$ $h'(x) = 2f'(x)\,(f(x) + f''(x))$ $c_1\in(-4,0),\ c_2\in(0,4):\ |f'(c_i)| \le 1 \Rightarrow h(c_i) \le 5$ $\max_{[c_1,c_2]} h \ge h(0) = 85 > 5 \Rightarrow \text{interior max at } \alpha$ $h'(\alpha)=0,\ f'(\alpha)^2 \ge 85-4 = 81 \ne 0 \Rightarrow f(\alpha)+f''(\alpha)=0$ $\boxed{(1),\ (2),\ (4) \text{ are TRUE}}$