(Solved): 5. [7 marks] Sketch a graph of a function \( f \) by hand that has all of the following propertie ...

5. [7 marks] Sketch a graph of a function \( f \) by hand that has all of the following properties: - \( f \) is continuous for all \( x \neq-3,1 \) - \( \lim _{x \rightarrow \infty} f(x)=-2 \) and \( \lim _{x \rightarrow-\infty} f(x)=-2 \) - \( f^{\prime}(x)=0 \) at \( x=0,3 \) only - \( f^{\prime}(x)>0 \) on \( (0,1) \cup(1,3) \) - \( f^{\prime}(x)<0 \) on \( (-\infty,-3) \cup(-3,0) \cup(3, \infty) \) - \( f^{\prime \prime}(x)=0 \) at \( x=5 \) only - \( \lim _{x \rightarrow-3^{-}} f(x) \rightarrow-\infty . \lim _{x \rightarrow-3^{+}} f(x) \rightarrow \infty, \lim _{x \rightarrow 1^{-}} f(x) \rightarrow \infty, \lim _{x \rightarrow 1^{+}} f(x) \rightarrow-\infty \)