Question

1. the table above gives selected values of the polynomial function f(x). assuming all zeroes are given, between what two points on the table will f(x) have a point of inflection? justify your answer.

x -3 0 3 6 9
p(x) -21 0 3 0 11

Explanation:

Step1: Recall inflection - point property

A point of inflection occurs where the second - derivative $f''(x)$ changes sign. This is related to the concavity of the function. We can use the fact that for a polynomial function, the sign change of the second - derivative can be inferred from the change in the rate of change of the first - derivative. We can estimate the first - derivative using the difference quotient $\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Calculate average rate of change between points

Between $x=-3$ and $x = 0$: $\frac{p(0)-p(-3)}{0-(-3)}=\frac{0 - (-21)}{3}=\frac{21}{3}=7$.
Between $x = 0$ and $x = 3$: $\frac{p(3)-p(0)}{3 - 0}=\frac{3-0}{3}=1$.
Between $x = 3$ and $x = 6$: $\frac{p(6)-p(3)}{6 - 3}=\frac{0 - 3}{3}=-1$.
Between $x = 6$ and $x = 9$: $\frac{p(9)-p(6)}{9 - 6}=\frac{11-0}{3}=\frac{11}{3}$.

Step3: Analyze the change in the rate of change

The average rate of change is decreasing from $x=-3$ to $x = 3$ (from $7$ to $1$) and then from $x = 3$ to $x = 6$ (from $1$ to $-1$), and then increasing from $x = 6$ to $x = 9$ (from $-1$ to $\frac{11}{3}$). The concavity changes from concave down (since the rate of change is decreasing) to concave up (since the rate of change is increasing). The change in concavity occurs between $x = 3$ and $x = 6$.

Answer:

The function $f(x)$ has a point of inflection between $x = 3$ and $x = 6$ because the average rate of change of the function changes from decreasing to increasing in this interval, which implies a change in concavity.