the given function f(x) is defined on the interval -9, 9 and represented by the graph below. for the circled point, select all that apply.

answer attempt 1 out of 2
the point is an absolute maximum
the point is an absolute minimum
the point is a relative maximum
the point is a relative minimum
the point is neither an absolute nor a relative extremum

Question

the given function f(x) is defined on the interval -9, 9 and represented by the graph below. for the circled point, select all that apply.

answer attempt 1 out of 2
the point is an absolute maximum
the point is an absolute minimum
the point is a relative maximum
the point is a relative minimum
the point is neither an absolute nor a relative extremum

Accepted Answer

# Brief Explanations:
1. **Absolute Maximum**: The absolute maximum is the highest point on the entire interval $[-9,9]$. The circled point is at $y = 0$, but there are points above this (e.g., the leftmost point has a higher $y$-value), so it's not an absolute maximum.
2. **Absolute Minimum**: The absolute minimum is the lowest point on $[-9,9]$. The circled point is at $y = 0$, and there are points below this (e.g., the two valley points have lower $y$-values), so it's not an absolute minimum.
3. **Relative Maximum**: A relative maximum is a point where the function changes from increasing to decreasing (a peak in its neighborhood). Looking at the graph, around the circled point, the function comes from a valley (increasing to the circled point) and then decreases after? Wait, no—wait, the left side: before the circled point, the function was decreasing to a valley, then increasing to the circled point, and then decreasing after? Wait, no, let's re-examine. The left part: from the leftmost point (high $y$), it decreases to a valley, then increases to the circled point (at $x$ near 0, $y=0$), then decreases to another valley, then increases again. Wait, a relative maximum is a point where $f(x)$ is greater than its immediate neighbors. At the circled point, the points to the left (immediate left) are lower (since it came from a valley, increasing to the circled point), and the points to the right (immediate right) are lower (since it decreases after the circled point). Wait, no—if it's increasing to the circled point and then decreasing, then the circled point would be a relative maximum. Wait, but the $y$-value is 0. Wait, maybe I misread. Wait, the graph: leftmost point (high $y$), then a line down to a valley (low $y$), then a line up to the circled point (at $x$ near 0, $y=0$), then a line down to another valley (low $y$), then a curve up to the rightmost point (at $y=0$). Wait, so at the circled point, the function was increasing (from the left valley) to the circled point, then decreasing (to the right valley). So in its neighborhood, the circled point is higher than its immediate left and right. So is it a relative maximum? Wait, but the $y$-value is 0. Wait, maybe the options: let's check the options again. Wait, the options are: absolute max, absolute min, relative max, relative min, neither. Wait, maybe I made a mistake. Wait, let's think again. A relative maximum is a local peak. So if the function increases to the point and then decreases, it's a relative max. But in this case, the circled point: left side (immediate left) is a valley (lower $y$), so the function is increasing to the circled point, then decreasing after. So the circled point is a local peak (relative max). But wait, the $y$-value is 0. Alternatively, maybe it's a relative minimum? No, relative minimum is a local valley. Wait, no—if it's increasing to the point and then decreasing, it's a max. If decreasing to the point and then increasing, it's a min. So here, increasing to the circled point (from left valley) and then decreasing (to right valley), so it's a relative maximum. But wait, the absolute max is the leftmost point (higher $y$), so not absolute. Absolute min is the two valleys (lower $y$). So relative max? Wait, but maybe the circled point is a relative maximum? Wait, but let's check the options. Wait, maybe the correct answer is "The point is a relative maximum" and "neither"? No, no—wait, maybe I messed up. Wait, let's look at the graph again (as per the description). The function is defined on $[-9,9]$. The circled point is at $x$ near 0, $y=0$. The leftmost point: higher $y$ (so absolute max is leftmost, not circled). The two valleys: lower $y$ (so absolute min is those, not circled). Now, relative max: in the neighborhood of the circled point, is $f(x)$ greater than its neighbors? To the left (immediate left), the function was at a valley (lower $y$), so it's increasing to the circled point. To the right (immediate right), the function decreases to another valley (lower $y$). So yes, the circled point is a relative maximum. Wait, but the $y$-value is 0. Alternatively, maybe the circled point is a relative maximum. But wait, the options: let's check the options. Wait, the user's options are:

- The point is an absolute maximum
- The point is an absolute minimum
- The point is a relative maximum
- The point is a relative minimum
- The point is neither an absolute nor a relative extremum

Wait, maybe I was wrong. Let's think about relative extrema. A relative maximum occurs when $f(c) \geq f(x)$ for $x$ near $c$. A relative minimum occurs when $f(c) \leq f(x)$ for $x$ near $c$. At the circled point $c$, take a small interval around $c$. To the left of $c$ (close to $c$), the function values are less than $f(c)$ (since it's increasing to $c$). To the right of $c$ (close to $c$), the function values are less than $f(c)$ (since it's decreasing after $c$). So $f(c)$ is greater than its neighbors, so it's a relative maximum. But wait, is it also an absolute maximum? No, because the leftmost point has a higher $f(x)$ value. So the circled point is a relative maximum, not absolute. Now, is it a relative minimum? No, because it's greater than neighbors. So the correct options would be "The point is a relative maximum" and? Wait, but maybe I made a mistake. Wait, maybe the graph is different. Wait, the user's graph: leftmost point (high $y$), then a line down to a valley (low $y$), then a line up to the circled point (at $x$ near 0, $y=0$), then a line down to another valley (low $y$), then a curve up to the rightmost point (at $y=0$). So at the circled point, the function is at a peak in its local neighborhood (since left is lower, right is lower). So it's a relative maximum. But wait, the $y$-value is 0. Alternatively, maybe the circled point is at $x=0$, $y=0$, and the left valley is at lower $y$, right valley at lower $y$, so yes, relative max. But wait, let's check the options again. Wait, maybe the answer is "The point is a relative maximum" and "neither" is wrong. Wait, no—maybe I messed up. Wait, let's check the definitions again. Absolute maximum: the highest point on the entire interval. The leftmost point is higher than the circled point, so not absolute max. Absolute minimum: the lowest points are the two valleys, so not absolute min. Relative maximum: yes, because it's a peak in its neighborhood. Relative minimum: no. So the correct option is "The point is a relative maximum". Wait, but maybe the graph is different. Wait, maybe the circled point is at the origin, and the function comes from below (left) and goes below (right), so it's a relative maximum. Alternatively, maybe the answer is "The point is a relative maximum" and "neither" is incorrect. Wait, maybe I was wrong. Let's think again. If the function is increasing to the circled point and then decreasing, then it's a relative maximum. So the correct option is "The point is a relative maximum". But wait, maybe the answer is "The point is a relative maximum" and also, is there another? Wait, the options are to select all that apply. So let's re-express:

- Absolute max: No (leftmost point is higher).
- Absolute min: No (valleys are lower).
- Relative max: Yes (local peak).
- Relative min: No (local peak, not valley).
- Neither: No (it is a relative max).

Wait, but maybe I made a mistake. Alternatively, maybe the circled point is a relative maximum. So the correct option is "The point is a relative maximum".
# Answer:
The point is a relative maximum

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