Question

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 the concept of concavity and inflection point

The rate of change of a function $g(x)$ is its first - derivative $g'(x)$. If the rate of change of $g$ (i.e., $g'(x)$) is increasing, then $g''(x)>0$ (the function is concave up). If the rate of change of $g$ (i.e., $g'(x)$) is decreasing, then $g''(x)<0$ (the function is concave down). An inflection point occurs where the second - derivative $g''(x)$ changes sign.

Step2: Analyze the given conditions

We are given that the rate of change of $g$ is increasing for $x < 2$ and decreasing for $x>2$. So, $g''(x)>0$ for $x < 2$ (the graph of $g$ is concave up for $x < 2$) and $g''(x)<0$ for $x>2$ (the graph of $g$ is concave down for $x > 2$). Since the concavity of the graph of $g$ changes at $x = 2$, the graph of $g$ 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 $x>2$.