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 ghas a minimum at x=2.
B. The graph of g has a maximum at x=2.
C. The graph of ghas a point of inflection at x=2 , is concave down for x<2, and is concave up for x>2.
D. The graph of ghas a point of inflection at x=2 , is concave up for x<2, and is concave down for x>2.

Answer 1

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.

Step-by-step explanation:

The correct answer is 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.

When the rate of change of a polynomial function is increasing for x<2 and decreasing for x>2, it indicates a change in concavity at x=2. This means that the function has a point of inflection at x=2.

Moreover, since the rate of change is increasing for x<2, it implies that the function is concave up for x<2. Conversely, since the rate of change is decreasing for x>2, it implies that the function is concave down for x>2.