Question

1. the rates of change of the polynomial function $g(x)$ are increasing for $x < - 3$ and decreasing for $x > - 3$. use this information to answer the questions below

a. on what interval is $g(x)$ concave up? explain how you know.
b. on what interval is $g(x)$ concave down? explain how you know.
c. what can you conclude about the point at $x=-3$? justify your answer.

Explanation:

Step1: Recall concavity - rate - of - change relationship

The second - derivative of a function $g(x)$ determines its concavity. If the rate of change of $g(x)$ (the first - derivative $g'(x)$) is increasing, then $g''(x)>0$ and the function is concave up. If the rate of change of $g(x)$ is decreasing, then $g''(x)<0$ and the function is concave down.

Step2: Determine concave - up interval

Since the rate of change of $g(x)$ (i.e., $g'(x)$) is increasing for $x < - 3$, then $g''(x)>0$ for $x < - 3$. So, $g(x)$ is concave up on the interval $(-\infty,-3)$.

Step3: Determine concave - down interval

Since the rate of change of $g(x)$ (i.e., $g'(x)$) is decreasing for $x > - 3$, then $g''(x)<0$ for $x > - 3$. So, $g(x)$ is concave down on the interval $(-3,\infty)$.

Step4: Analyze the point at $x=-3$

The concavity of $g(x)$ changes at $x = - 3$ (from concave up to concave down). By the definition of an inflection point, a point where the concavity of a function changes is an inflection point. So, the point $x=-3$ is an inflection point of the function $g(x)$.

Answer:

A. The function $g(x)$ is concave up on the interval $(-\infty,-3)$ because the rate of change of $g(x)$ (the first - derivative $g'(x)$) is increasing for $x < - 3$, which means $g''(x)>0$ on this interval.
B. The function $g(x)$ is concave down on the interval $(-3,\infty)$ because the rate of change of $g(x)$ (the first - derivative $g'(x)$) is decreasing for $x > - 3$, which means $g''(x)<0$ on this interval.
C. The point $x = - 3$ is an inflection point of the function $g(x)$ since the concavity of $g(x)$ changes from concave up (for $x < - 3$) to concave down (for $x > - 3$) at $x=-3$.