Question 79
Suppose a quadratic function $f(x) = a x^2 + b x + c$ ($a, b, c \in \mathbb{R}$, and $a \neq 0$) satisfies the following conditions:
(1) When $x \in \mathbb{R}$, $f(x-4) = f(2-x)$ and $f(x) \ge x$.
(2) When $x \in (0, 2)$, $f(x) \le \left(\frac{x+1}{2}\right)^2$.
(3) The minimum value of $f(x)$ on $\mathbb{R}$ is $0$.
Find the maximal $m$ ($m \gt 1$) such that there exists $t \in \mathbb{R}$, $f(x+t) \le x$ holds so long as $x \in [1, m]$.
(1) When $x \in \mathbb{R}$, $f(x-4) = f(2-x)$ and $f(x) \ge x$.
(2) When $x \in (0, 2)$, $f(x) \le \left(\frac{x+1}{2}\right)^2$.
(3) The minimum value of $f(x)$ on $\mathbb{R}$ is $0$.
Find the maximal $m$ ($m \gt 1$) such that there exists $t \in \mathbb{R}$, $f(x+t) \le x$ holds so long as $x \in [1, m]$.