A function $f(x)$ is continuous on the set of all real numbers, $f(x) \geq 0$ for all real numbers $x$, and $f(x) = -4xe^{4x^2}$ for $x < 0$. For all positive numbers $t$, the equation $f(x) = t$ has exactly 2 distinct real roots. Let $g(t)$ denote the smaller root and $h(t)$ denote the larger root of this equation. The two functions $g(t)$ and $h(t)$ satisfy $$2g(t) + h(t) = k \quad (k \text{ is a constant})$$ for all positive numbers $t$. If $\int_0^7 f(x)\,dx = e^4 - 1$, find the value of $\frac{f(9)}{f(8)}$. [4 points] (1) $\frac{3}{2}e^5$ (2) $\frac{4}{3}e^7$ (3) $\frac{5}{4}e^9$ (4) $\frac{6}{5}e^{11}$ (5) $\frac{7}{6}e^{13}$
A function $f(x)$ is continuous on the set of all real numbers, $f(x) \geq 0$ for all real numbers $x$, and $f(x) = -4xe^{4x^2}$ for $x < 0$.\\
For all positive numbers $t$, the equation $f(x) = t$ has exactly 2 distinct real roots. Let $g(t)$ denote the smaller root and $h(t)$ denote the larger root of this equation.\\
The two functions $g(t)$ and $h(t)$ satisfy
$$2g(t) + h(t) = k \quad (k \text{ is a constant})$$
for all positive numbers $t$. If $\int_0^7 f(x)\,dx = e^4 - 1$, find the value of $\frac{f(9)}{f(8)}$. [4 points]\\
(1) $\frac{3}{2}e^5$\\
(2) $\frac{4}{3}e^7$\\
(3) $\frac{5}{4}e^9$\\
(4) $\frac{6}{5}e^{11}$\\
(5) $\frac{7}{6}e^{13}$