Question

example 2: the figure shows the graph of a function g in the xy - plane with four labeled points. of the following points, at which is the rate of change of g the least? (a) a (b) b (c) c (d) d

Explanation:

Step1: Understand Rate of Change

The rate of change of a function at a point is related to the slope of the tangent line at that point. A more negative slope means a smaller (more negative) rate of change.

Step2: Analyze Each Point

• Point A: It's a local minimum, so the tangent slope is 0 (horizontal tangent).
• Point B: The function is increasing here, so the slope is positive.
• Point C: The function is still increasing but less steeply than at B, or maybe starting to decrease? Wait, no, looking at the graph, from the peak towards C, then D. Wait, actually, at point D, the graph is decreasing very steeply. Wait, let's re-examine:
• Point A: local min, slope 0.
• Point B: on the increasing part, slope positive.
• Point C: on the decreasing part but less steep? Wait, no, the graph goes up to a peak, then down. So after the peak, at C, then D. The slope at D is more negative (steeper downward) than at C.
• Wait, the question is "the rate of change of \( g \) the least". Since rate of change can be negative (decreasing), a more negative rate is smaller (lesser). So we need the point with the most negative slope (steepest downward).

Looking at the points:

• A: slope 0 (horizontal)
• B: slope positive (increasing)
• C: slope negative but less steep (mildly decreasing)
• D: slope negative and very steep (strongly decreasing)

So the rate of change at D is the least (most negative).

Answer:

D