QUESTION IMAGE
Question
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.
| x | f(x) | f(x) | |
|---|---|---|---|
| x = -2 | |||
| -2 < x < 0 | |||
| x = 0 | |||
| 0 < x < 2 | |||
| x = 2 | |||
| 2 < x < ∞ |
sketch the graph of f(x). choose the correct graph
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
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
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
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
If \(f^{\prime}(x)\) is negative and decreasing in \(2 < x<\infty\), then \(f(x)\) is decreasing and concave - down.
The completed table:
\(x\) \(f^{\prime}(x)\) \(f(x)\) \(x=-2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(-2Positive and increasing Increasing and concave - up \(x = 0\) Zero Horizontal tangent \(0Negative and increasing Decreasing and concave - up \(x = 2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(2Negative 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 \(x\in(2,\infty)\).
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
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
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
If \(f^{\prime}(x)\) is negative and decreasing in \(2 < x<\infty\), then \(f(x)\) is decreasing and concave - down.
The completed table:
\(x\) \(f^{\prime}(x)\) \(f(x)\) \(x=-2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(-2Positive and increasing Increasing and concave - up \(x = 0\) Zero Horizontal tangent \(0Negative and increasing Decreasing and concave - up \(x = 2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(2Negative 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 \(x\in(2,\infty)\).
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
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
If \(f^{\prime}(x)\) is negative and decreasing in \(2 < x<\infty\), then \(f(x)\) is decreasing and concave - down.
The completed table:
\(x\) \(f^{\prime}(x)\) \(f(x)\) \(x=-2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(-2Positive and increasing Increasing and concave - up \(x = 0\) Zero Horizontal tangent \(0Negative and increasing Decreasing and concave - up \(x = 2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(2Negative 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 \(x\in(2,\infty)\).
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
If \(f^{\prime}(x)\) is negative and decreasing in \(2 < x<\infty\), then \(f(x)\) is decreasing and concave - down.
The completed table:
\(x\) \(f^{\prime}(x)\) \(f(x)\) \(x=-2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(-2Positive and increasing Increasing and concave - up \(x = 0\) Zero Horizontal tangent \(0Negative and increasing Decreasing and concave - up \(x = 2\) Local maximum of \(f^{\prime}(x)\) Inflection point \(2Negative 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 \(x\in(2,\infty)\).
If \(f^{\prime}(x)\) is negative and decreasing in \(2 < x<\infty\), then \(f(x)\) is decreasing and concave - down.
The completed table:
| \(x\) | \(f^{\prime}(x)\) | \(f(x)\) |
|---|---|---|
| \(x=-2\) | Local maximum of \(f^{\prime}(x)\) | Inflection point |
\(-2| Positive and increasing | Increasing and concave - up | |
| \(x = 0\) | Zero | Horizontal tangent |
\(0| Negative and increasing | Decreasing and concave - up | |
| \(x = 2\) | Local maximum of \(f^{\prime}(x)\) | Inflection point |
\(2| 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 \(x\in(2,\infty)\).
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| \(x\) | \(f^{\prime}(x)\) | \(f(x)\) |
|---|---|---|
| \(x=-2\) | Local maximum of \(f^{\prime}(x)\) | Inflection point |
\(-2| Positive and increasing | Increasing and concave - up | |
| \(x = 0\) | Zero | Horizontal tangent |
\(0| Negative and increasing | Decreasing and concave - up | |
| \(x = 2\) | Local maximum of \(f^{\prime}(x)\) | Inflection point |
\(2| Negative and decreasing | Decreasing and concave - down | |
Sketch as described above.