Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options. (1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$ (2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$ (3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$ (4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$ (5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$
Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options.\\
(1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$\\
(2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$\\
(3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$\\
(4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$\\
(5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$