the function $f(x)$ is graphed below. how many points on the graph represent a relative maximum?
graph of a function with points labeled a, b, c, d, e
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# Explanation:
## Step1: Define relative maximum
A relative maximum is a point where the function changes from increasing to decreasing, and the function value at that point is greater than the values at neighboring points.

## Step2: Analyze each point
- Point $a$: It's an $x$-intercept, the function is decreasing before and increasing after? No, before $a$ the function is going up, after $a$ it goes down to $b$, but $a$ is on the $x$-axis, not a peak.
- Point $b$: It's a relative minimum (function changes from decreasing to increasing here).
- Point $c$: $x$-intercept, function is increasing before and after? No, before $c$ it was increasing from $b$, after $c$ it continues to increase to $d$.
- Point $d$: Here, the function changes from increasing (coming from $c$ to $d$) to decreasing (after $d$ it goes down), and $d$ is higher than its neighbors. So $d$ is a relative maximum.
- Point $e$: It's a point where the function touches the $x$-axis and then goes down, but before $e$ the function was decreasing to $e$ (from $d$ to $e$ it's decreasing) and after $e$ it's decreasing more. So $e$ is not a relative maximum (it's a point where the function stops decreasing? Wait, no, the graph approaches $e$ from above (since before $e$ the function is above $x$-axis, touches $e$ and then goes down). Wait, actually, at $e$, the function has a horizontal tangent? But the key is relative maximum: a point where around it, the function is higher. At $d$, the function is higher than left (from $c$ to $d$ increasing) and right (from $d$ to $e$ decreasing). At $e$, the left side (towards $d$) is decreasing to $e$, and right side is decreasing from $e$. So $e$ is not a relative maximum. Only $d$ is a relative maximum? Wait, wait, maybe I made a mistake. Wait, the graph: from $b$ to $d$, the function is increasing (from $b$ up to $d$), then from $d$ to $e$ it's decreasing, then from $e$ down. So $d$ is a relative maximum. Is there another? Wait, the left part: from the left end (going down to $a$, then down to $b$, then up to $c$, then up to $d$). Wait, $b$ is a minimum. $d$ is a maximum. What about the leftmost part? The left end is going up, so as $x$ approaches $-\infty$, the function goes to $+\infty$? Wait, no, the left end arrow is going up, so the function is increasing as $x$ approaches $-\infty$? Wait, the graph: leftmost point, the arrow is up, so when $x$ is very small (negative, large magnitude), $f(x)$ is large positive, then it comes down to $a$, then down to $b$, then up to $c$, then up to $d$, then down to $e$, then down. Wait, so before $a$, the function is decreasing (from $+\infty$ to $a$), then from $a$ to $b$ it's decreasing, then from $b$ to $d$ it's increasing (since $b$ to $c$ to $d$ is up), then $d$ to $e$ is down, $e$ to right is down. So the relative maximum: a point where the function changes from increasing to decreasing. So from $b$ to $d$, the function is increasing (so the slope is positive), then from $d$ to $e$, slope is negative. So $d$ is a relative maximum. Is there another? Wait, the leftmost part: as $x$ approaches $-\infty$, $f(x)$ approaches $+\infty$, but that's a limit, not a point. The actual points: $d$ is the only point where the function changes from increasing to decreasing. Wait, but wait, maybe I misread the graph. Let me re-examine: the graph has a minimum at $b$, then increases to $d$, then decreases to $e$, then decreases more. So $d$ is the only relative maximum. Wait, but the question is "how many points on the graph represent a relative maximum". So only $d$? Wait, no, wait, maybe the left end: the function comes from the top left (going down to $a$, then down to $b$, then up to $c$, then up to $d$). Wait, no, the left arrow is going up, so when $x$ is very negative, $f(x)$ is large, then it decreases to $a$, then decreases to $b$, then increases to $c$, then increases to $d$, then decreases to $e$, then decreases. So the function has a relative maximum at $d$ (since it goes up to $d$ then down) and is there another? Wait, the leftmost part: the function is decreasing from $+\infty$ (as $x\to -\infty$) to $b$? No, wait, the graph: starting from the left (x very negative), the function is at a high value (arrow up), then it comes down (decreasing) to point $a$ (on x-axis), then continues decreasing to point $b$ (a minimum), then starts increasing (from $b$ to $c$ to $d$), reaches $d$ (a peak), then decreases to $e$ (on x-axis), then decreases further. So the only point where the function changes from increasing to decreasing is $d$. So there's 1 relative maximum.

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Explanation:

Step1: Define relative maximum

Step2: Analyze each point

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