Question

which of the following is most likely to be a relative minimum for this graph?
graph with points (-1, 0), (2, 0), (4, 0) on x-axis, and a red curve
a. (4, 0)
b. (-1, 0)
c. (0.5, 8.1)
d. (3, -4)

Explanation:

Step1: Recall relative minimum definition

A relative minimum is a point where the function changes from decreasing to increasing, and the \( y \)-value is lower than its neighboring points.

Step2: Analyze each option

• Option A: \((4, 0)\) is an \( x \)-intercept, not a minimum (neighboring points around \( x = 4 \) have higher or same \( y \)-values? Wait, no—look at the graph: the curve near \( x = 4 \) comes from below? Wait, no, the graph has a minimum between \( x = 2 \) and \( x = 4 \). Wait, let's check each:
• Option B: \((-1, 0)\) is an \( x \)-intercept, the function is increasing before and after? No, the left side of \( x = -1 \) is going down, right side (towards \( x = 0 \)) is going up? Wait, no, the graph at \( x = -1 \) is a root, and the function near \( x = -1 \): left of \( x = -1 \), the graph is going down (since it's a vertical asymptote? Wait, no, the left side is a vertical line? Wait, no, the graph has a vertical asymptote? Wait, no, the left part is a curve? Wait, the graph has roots at \( x = -1 \), \( x = 2 \), \( x = 4 \). The peak at \( (0.5, 8.1) \) (option C) is a relative maximum. Then between \( x = 2 \) and \( x = 4 \), the graph has a valley (relative minimum). Option D: \((3, -4)\) is in that interval, with \( y = -4 \), which is lower than neighbors. Options A and B are roots ( \( y = 0 \) ), higher than \( -4 \). Option C is a maximum.

Answer:

D. \((3, -4)\)