identify (a) the point(s) of inflection and (b) the intervals where the function is concave up or concave…

Explanation:

Step1: Recall inflection - point definition

Inflection points are where the concavity of the function changes. Visually, it is where the curve changes from concave - up to concave - down or vice - versa.

Step2: Analyze the graph

Looking at the graph, the curve changes concavity at the point $(0,-1)$.

Step3: Recall concavity intervals

A function is concave up when the second - derivative is positive and concave down when the second - derivative is negative. Visually, a function is concave up if it "holds water" and concave down if it "spills water".

Step4: Determine concavity intervals from the graph

The function is concave down on the interval $(-\infty,0)$ and concave up on the interval $(0,\infty)$.

Answer:

D. (a) Inflection point at $(0, - 1)$, (b) concave upward on $(0,\infty)$, concave downward on $(-\infty,0)$