To determine where $p(x)$ has an inflection point, we need to look at the values in the table for $p''(x)$. An inflection point occurs when the second derivative, $p''(x)$, changes sign, indicating a change in the concavity of $p(x)$.

From the table:

We observe the sign changes in $p''(x)$ between consecutive values:

• Between $x = 0$ and $x = 1$, $p''(x)$ changes from $0$ to $2$ (indicating a shift from zero to positive values).

Therefore, there is a sign change in $p''(x)$ around $x = 0$ and $x = 1$. Based on this table and continuity assumptions, $p(x)$ must have an inflection point at $x = 0$.

Would you like further details on this explanation or additional questions to explore related concepts?

Here are five related questions to deepen understanding:

1. What conditions are necessary for a function to have a local maximum or minimum?
2. How does the sign of the first derivative $p'(x)$ relate to the increasing or decreasing behavior of $p(x)$?
3. Can a point where $p''(x) = 0$ always be considered an inflection point? Why or why not?
4. What additional information would be required to determine the specific shape of $p(x)$ between the given values?
5. How does continuity of $p(x)$, $p'(x)$, and $p''(x)$ affect the identification of critical points and inflection points?

Tip: For a function $p(x)$, always examine the signs of $p'(x)$ and $p''(x)$ in addition to their values to identify points of interest such as extrema and inflection points.