Sketch function from derivatives and limits

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice-differentiable function satisfying the conditions below:

• As $x \to -\infty$, $f(x) \to 0$, and as $x \to +\infty$, $f(x) \to 5$.
• The derivative is positive on $(-\infty, -2)$ and $(1, 3)$, negative on $(-2, 1)$ and $(3, \infty)$, and zero only at $x = -2, 1, 3$.
• The second derivative is negative on $(-\infty, -1)$ and $(-1, 2)$, positive on $(2, \infty)$, and undefined at $x = 2$; however, $f$ itself is continuous everywhere.

Tasks: (a) Draw a qualitatively accurate graph of $f$, labeling every local extremum and any inflection point. (b) Can $f$ have two local maxima and one local minimum while satisfying these conditions? Give a rigorous justification. (c) Identify where an inflection point is forced to occur, if one exists, and describe the sign of the slope just to the left and just to the right of $x = 2$.

Overview: The problem assesses grasp of core calculus ideas—derivatives, concavity, limits, and continuity—and the skill of translating derivative sign patterns into conclusions about monotonicity and local extrema.