Question: Absolute Maximum is ____ at x = none
Problems 18 - 21, The graph of g(x) is shown below. The points of inflection are labeled as A and B.
Find the open intervals where the rate of change of g(x) has the following
properties.
Graph of g(x)
18. The rate of change of g(x) is positive and increasing.
19. The rate of change of g(x) is positive and decreasing.
20. The rate of change of g(x) is negative and increasing.
21. The rate of change of g(x) is negative and decreasing.
Graph shape: a smooth curve on an x-y coordinate plane with labeled points C near the top-left local maximum around $(-3, 8)$, A on the descending branch left of the y-axis, D near the local minimum around $(2, 1)$, B on the rising branch around $(3, 2)$, and F near the smaller local maximum around $(4, 3)$; the curve crosses the x-axis near $x = -5$ and $x = 6$ and continues downward at both ends. position_hint=center
1. **State the problem:** We need to find the open intervals where the rate of change of $g(x)$ (i.e., $g'(x)$) is positive and increasing, positive and decreasing, negative and increasing, and negative and decreasing.
2. **Recall definitions:**
- The rate of change of $g(x)$ is $g'(x)$.
- $g'(x)$ positive means $g(x)$ is increasing.
- $g'(x)$ negative means $g(x)$ is decreasing.
- The rate of change of $g'(x)$ is $g''(x)$ (the second derivative).
- $g'(x)$ increasing means $g''(x) > 0$.
- $g'(x)$ decreasing means $g''(x) < 0$.
3. **Analyze intervals using points of inflection $A$ and $B$:**
- Points $A$ and $B$ are inflection points where $g''(x) = 0$ and the concavity changes.
4. **Intervals for each property:**
**18. $g'(x)$ positive and increasing:**
- $g'(x) > 0$ and $g''(x) > 0$
- From the graph, $g'(x) > 0$ roughly between $x = -5$ and $x = -3$ (increasing to local max at $C$), and between $x = 2$ and $x = 6$ (rising branch).
- $g''(x) > 0$ between $A$ and $B$ (inflection points).
- So, positive and increasing occurs on $(A, B)$ where $g'(x) > 0$.
- From graph, this is approximately $(3, 6)$.
**19. $g'(x)$ positive and decreasing:**
- $g'(x) > 0$ and $g''(x) < 0$
- This occurs between $x = -5$ and $A$ (left of $A$), where $g'(x)$ is positive but concavity is downward.
- Approximate interval: $(-5, 3)$ but only where $g'(x) > 0$, so roughly $(-5, -3)$.
**20. $g'(x)$ negative and increasing:**
- $g'(x) < 0$ and $g''(x) > 0$
- This occurs between $B$ and $6$ where $g'(x)$ is negative but concavity is upward.
- Approximate interval: $(3, 6)$ but $g'(x)$ is positive here, so no.
- Check between $-3$ and $2$ (around $D$), $g'(x)$ is negative and $g''(x) > 0$ between $A$ and $B$?
- Actually, between $-3$ and $2$, $g'(x)$ is negative and $g''(x) > 0$ between $A$ and $B$.
- So interval is approximately $( -3, 2 )$.
**21. $g'(x)$ negative and decreasing:**
- $g'(x) < 0$ and $g''(x) < 0$
- This occurs left of $A$ and right of $B$ where $g'(x)$ is negative and concavity downward.
- Approximate intervals: $(- ext{infinity}, A)$ and $(B, ext{infinity})$ where $g'(x) < 0$.
- From graph, $g'(x)$ negative and decreasing roughly on $(- ext{infinity}, -5)$ and $(2, 3)$.
5. **Summary of intervals:**
- 18: $g'(x)$ positive and increasing on approximately $(3, 6)$.
- 19: $g'(x)$ positive and decreasing on approximately $(-5, -3)$.
- 20: $g'(x)$ negative and increasing on approximately $(-3, 2)$.
- 21: $g'(x)$ negative and decreasing on approximately $(- ext{infinity}, -5)$ and $(2, 3)$.
6. **Final answers:**
- 18. $(3, 6)$
- 19. $(-5, -3)$
- 20. $(-3, 2)$
- 21. $(- ext{infinity}, -5) \cup (2, 3)$