Problem Analysis #33

Let $f(x)$ be a cubic function whose leading coefficient is positive, and let $t$ be a real number. Define$g(x) = \begin{cases}-f(x) & (x < t), \\[4pt]f(x) & (x \geq t),\end{cases}$so that $g$ is continuous on the entire real line and satisfies the following conditions.\begin{center}\fbox{\begin{minipage}{0.9\textwidth}(A) For every real number $a$, the limit$\displaystyle\lim_{x \to a^+} \dfrac{g(x)}{x(x-2)}$exists.KOR-CSAT-(B) The set of natural numbers $m$ for which$\displaystyle\lim_{x \to m^+} \dfrac{g(x)}{x(x-2)}$is negative is$\left\{g(-1), -\dfrac{7}{2}g(1)\right\}.$\end{minipage}}\end{center}Assuming that $g(-1) eq -\dfrac{7}{2}g(1)$, compute $g(-5)$. [4 points]\