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 concavity and rate of change

The rate of change of \( g \) is \( g'(x) \). The rate of change of \( g' \) (i.e., \( g''(x) \)) determines concavity. If \( g'(x) \) is increasing, \( g''(x)>0 \) (concave up). If \( g'(x) \) is decreasing, \( g''(x)<0 \) (concave down).

Step2: Analyze the given conditions

For \( x < 2 \), \( g' \) is increasing, so \( g''(x)>0 \) (concave up). For \( x > 2 \), \( g' \) is decreasing, so \( g''(x)<0 \) (concave down). A point where concavity changes is an inflection point, so at \( x = 2 \), there's an inflection point.

Step3: Evaluate options

• Option A: Minima/maxima relate to \( g'(x)=0 \), not \( g''(x) \) change. Eliminate.
• Option B: Same as A, relates to \( g'(x)=0 \). Eliminate.
• Option C: Concavity is up for \( x < 2 \), down for \( x > 2 \), not as stated. Eliminate.
• Option D: Matches the concavity (up for \( x < 2 \), down for \( x > 2 \)) and 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 \( x > 2 \)