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The x-intercepts are the x-values of the points where the graph intercepts the x-axis.
Graph:
x-intercepts: x≈ - 3.07
Increasing Interval: x ≤ - 1.72 and x ≥ 0.39
Decreasing Interval: - 1.72 ≤ x ≤ 0.39
Local Maximum: (-1.72, 4.13)
Local Minimum: (0.39, 1.79)
Let's first draw the graph of the function. Then we will use the graph to identify the increasing and decreasing intervals.
We want to find the x-intercepts, and the local minimum and maximum. Let's think about what these terms mean and then obtain the desired information.
x-intercepts | Relative minimum and maximum | |
---|---|---|
Definition | x-coordinates of the points at which the graph intersects the x-axis | The lowest and highest point for a region of the graph. |
In the Given Function | -3.07 | (-1.72, 4.13) and (0.39, 1.79) |
A local maximum is the point where the function goes from increasing interval to decreasing interval. A local minimum is the point where the function goes from decreasing interval to increasing interval.
As seen on the graph, the function has one local maximum and one local minimum. Local Maximum:& (-1.72, 4.13) Local Minimum:& (0.39, 1.79)
To determine the intervals for which the function is increasing, decreasing, and constant, let's consider the graph of the function as we move along it in the positive x-direction.
We will plug in x-values to get the y-values.
x | 0.5x^3+x^2-x+2 | y= 0.5x^3+x^2-x+2 |
---|---|---|
- 3 | 0.5( - 3)^3+( - 3)^2-( - 3)+2 | 0.5 |
- 2 | 0.5( - 2)^3+( - 2)^2-( - 2)+2 | 4 |
- 1 | 0.5( - 1)^3+( - 1)^2-( - 1)+2 | 3.5 |
0 | 0.5( 0)^3+( 0)^2-( 0)+2 | 2 |
1 | 0.5( 1)^3+( 1)^2-( 1)+2 | 2.5 |
We can plot these points, and connect them to get the graph of the function.