Question

which of the following is most likely to be a relative maximum for this graph? a. (0.5, -8.1) b. (-1, 0) c. (3, 4) d. (4, 0)

Explanation:

Step1: Recall relative maximum definition

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

Step2: Analyze each option

• Option A: \((0.5, -8.1)\) has a negative \( y \)-value. Looking at the graph, there's a lower point, but this seems like a minimum (since it's a low point) rather than a maximum.
• Option B: \((-1, 0)\) is an \( x \)-intercept. The graph is decreasing before \( x=-1 \) (from the left) and then? Wait, the graph near \( x=-1 \): the left side of \( x=-1 \) (like \( x < -1 \)) – the graph is above, and at \( x=-1 \), it's a root. But the function's behavior: before \( x=-1 \), it's decreasing? Wait, no, the graph on the left of \( x=-1 \) (like \( x=-2 \)) – the graph is high, then at \( x=-1 \), it's 0. Then after \( x=-1 \), it goes down. Wait, but a relative maximum should be a peak. \((-1, 0)\) is a root, not a peak.
• Option C: \((3, 4)\) – the \( y \)-value is positive (4). Looking at the graph, there's a peak between \( x=2 \) and \( x=4 \) (since the roots are at \( x=2 \), \( x=4 \)? Wait, the roots are at \( x=-1 \), \( x=2 \), \( x=4 \)? Wait, the graph: left side, a vertical asymptote? Wait, no, the graph has a root at \( x=-1 \), \( x=2 \), \( x=4 \). Then between \( x=2 \) and \( x=4 \), the graph has a peak. So \((3, 4)\) is inside that interval, and the \( y \)-value is positive, which is a peak (relative maximum) because around \( x=3 \), the function increases to \( x=3 \) then decreases.
• Option D: \((4, 0)\) is an \( x \)-intercept, a root, not a maximum.

Answer:

C. \((3, 4)\)