Question

select the correct answer. which statement best describes the graphed function? a. the function has a relative maximum at (-2,4) and a relative minimum at (2,-4). b. the function has a relative minimum at (-2,4) and a relative maximum at (2,-4). c. the function has an absolute maximum at (-2,4) and an absolute minimum at (2,-4). d. the function has an absolute minimum at (-2,4) and an absolute maximum at (2,-4).

Explanation:

To determine the correct statement, we analyze the graph:

1. Relative vs. Absolute Extrema:

• A relative maximum/minimum is a peak/valley in a local region. An absolute maximum/minimum is the highest/lowest point on the entire graph.
• The graph has a peak (relative maximum) at \((-2, 4)\) and a valley (relative minimum) at \((2, -4)\) because these are local high and low points.
• For absolute extrema, the graph extends infinitely (as seen from the ends approaching horizontal asymptotes), so there are no absolute maxima or minima (the function does not have a single highest or lowest point over all real numbers).

1. Analyzing Options:

• Option A: Correctly identifies a relative maximum at \((-2, 4)\) and a relative minimum at \((2, -4)\) (matches the local extrema).
• Option B: Swaps relative minimum and maximum (incorrect).
• Options C and D: Claim absolute extrema, but the graph has no absolute extrema (since it does not have a global highest or lowest point).

Answer:

A. The function has a relative maximum at (-2,4) and a relative minimum at (2,-4).