use the graph of y = f(x) to discuss the graph of y = f(x). organize your conclusions in a table, and sketch…
use the graph of y = f(x) to discuss the graph of y = f(x). organize your conclusions in a table, and sketch a possible graph of y = f(x). determine how the properties on different intervals of f(x) affect f(x). complete the table.\n\n| x | f(x) | f(x) |\n|----|----|----|----|\n| -∞ < x < -2 | positive and decreasing | | |\n| x = -2 | | | |\n| -2 < x < 0 | | | |\n| x = 0 | | | |\n| 0 < x < 2 | | | |\n| x = 2 | | | |\n| 2 < x < ∞ | | | |\nsketch the graph of f(x). choose the correct graph
Answer
Explanation:
Step1: Recall derivative - function relationship
If (f^{\prime}(x)>0), (f(x)) is increasing. If (f^{\prime}(x)<0), (f(x)) is decreasing. If (f^{\prime}(x)) is increasing, (f(x)) is concave - up and if (f^{\prime}(x)) is decreasing, (f(x)) is concave - down.
Step2: Analyze (-\infty<x < - 2)
Since (f^{\prime}(x)) is positive and decreasing, (f(x)) is increasing and concave - down.
Step3: Analyze (x=-2)
When (f^{\prime}(x)) changes its behavior (from positive - decreasing to other values), (f(x)) may have an inflection point. Here, (f^{\prime}(-2)) is a local maximum of (f^{\prime}(x)), so (f(x)) has an inflection point at (x = - 2).
Step4: Analyze (-2<x<0)
If (f^{\prime}(x)) is positive and increasing in (-2 < x<0), then (f(x)) is increasing and concave - up.
Step5: Analyze (x = 0)
If (f^{\prime}(0)=0), (f(x)) has a horizontal tangent at (x = 0).
Step6: Analyze (0<x<2)
If (f^{\prime}(x)) is negative and increasing in (0 < x<2), then (f(x)) is decreasing and concave - up.
Step7: Analyze (x = 2)
If (f^{\prime}(2)) is a local maximum of (f^{\prime}(x)) (change in the behavior of (f^{\prime}(x)) from increasing to decreasing), (f(x)) has an inflection point at (x = 2).
Step8: Analyze (2<x<\infty)
If (f^{\prime}(x)) is negative and decreasing in (2 < x<\infty), then (f(x)) is decreasing and concave - down.
The completed table:
| (x) | (f^{\prime}(x)) | (f(x)) |
|---|---|---|
| (-\infty<x < - 2) | Positive and decreasing | Increasing and concave - down |
| (x=-2) | Local maximum of (f^{\prime}(x)) | Inflection point |
| (-2<x<0) | Positive and increasing | Increasing and concave - up |
| (x = 0) | Zero | Horizontal tangent |
| (0<x<2) | Negative and increasing | Decreasing and concave - up |
| (x = 2) | Local maximum of (f^{\prime}(x)) | Inflection point |
| (2<x<\infty) | Negative and decreasing | Decreasing and concave - down |
To sketch the graph of (y = f(x)):
- Start with an increasing and concave - down curve for (x\in(-\infty,-2)).
- At (x=-2), the curve has an inflection point and then becomes increasing and concave - up until (x = 0) where it has a horizontal tangent.
- Then it becomes decreasing and concave - up until (x = 2) where it has another inflection point.
- After (x = 2), it is decreasing and concave - down for (x\in(2,\infty)).
Answer:
| (x) | (f^{\prime}(x)) | (f(x)) |
|---|---|---|
| (-\infty<x < - 2) | Positive and decreasing | Increasing and concave - down |
| (x=-2) | Local maximum of (f^{\prime}(x)) | Inflection point |
| (-2<x<0) | Positive and increasing | Increasing and concave - up |
| (x = 0) | Zero | Horizontal tangent |
| (0<x<2) | Negative and increasing | Decreasing and concave - up |
| (x = 2) | Local maximum of (f^{\prime}(x)) | Inflection point |
| (2<x<\infty) | Negative and decreasing | Decreasing and concave - down |
| Sketch as described above. |