Which of the following most accurately identifies the relative maximum and minimum of the polynomial function h(x)?
(1 point)
Responses
There is a relative maximum at x=−1 and a relative minimum at x=0.555.
There is a relative maximum at x equals negative 1 and a relative minimum at x equals 0.555 .
There is a relative maximum at x=1.5 and a relative minimum at x=−2.
There is a relative maximum at x equals 1.5 and a relative minimum at x equals negative 2 .
There is a relative maximum at x=−1 and no relative minimum.
There is a relative maximum at x equals negative 1 and no relative minimum.
There is a relative minimum at x=−1 and a relative maximum at x=0.555.
There is a relative minimum at x=−1 and a relative maximum at x=0.555.
The correct response is: "There is a relative maximum at x=−1 and a relative minimum at x=0.555."
To determine the relative maximum and minimum of a polynomial function, you need to analyze the function's graph. Here's how you can do that:
1. Plot the function on a graph: Start by plotting the polynomial function h(x) on a coordinate plane. This will give you a visual representation of the function's shape.
2. Identify turning points: Look for places where the graph changes from increasing to decreasing or vice versa. These points are called turning points and can be potential relative maximum or minimum points.
3. Determine the nature of the turning points: To determine whether a turning point is a relative maximum or minimum, you need to examine the behavior of the graph near the point.
- Relative maximum: If the graph is increasing on both sides of the turning point, it is a relative maximum.
- Relative minimum: If the graph is decreasing on both sides of the turning point, it is a relative minimum.
Based on the given options:
Option 1: The function has a relative maximum at x = -1 and a relative minimum at x = 0.555. This option provides the correct answer.
Option 2: The function has a relative maximum at x = 1.5 and a relative minimum at x = -2. This option does not match the given options.
Option 3: The function has a relative maximum at x = -1 and no relative minimum. This option does not match the given options.
Option 4: The function has a relative minimum at x = -1 and a relative maximum at x = 0.555. This option does not match the given options.
Therefore, the correct answer is: There is a relative maximum at x = -1 and a relative minimum at x = 0.555.