Question

1 mark for review for the polynomial function g, the rate of change of g is increasing for x < 2 and decreasing for x > 2. which of the following must be true? a the graph of g has a minimum at x = 2. b the graph of g has a maximum at x = 2. c the graph of g has a point of inflection at x = 2, is concave down for x < 2, and is concave up for x > 2. d the graph of g has a point of inflection at x = 2, is concave up for x < 2, and is concave down for x > 2.

Explanation:

Step1: Recall concavity and rate - of - change relationship

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

Step2: Analyze the given information

We are given that the rate of change of $g$ (i.e., $g'(x)$) is increasing for $x < 2$ and decreasing for $x>2$. So, $g''(x)>0$ for $x < 2$ (concave up) and $g''(x)<0$ for $x>2$ (concave down). A point of inflection occurs where the concavity changes. Since the concavity changes at $x = 2$, the function $g(x)$ has a point of inflection at $x = 2$.

Answer:

D. The graph of $g$ has a point of inflection at $x = 2$, is concave up for $x < 2$, and is concave down for $x>2$