Question

the function $f(x)$ is graphed below. how many points on the graph represent a relative extreme value?
(graph of a function with points a, b, c, d, e: a and e are x-intercepts, b is a local maximum, d is a local minimum, c is an x-intercept between b and d)

Explanation:

Step1: Define relative extreme value

A relative extreme value (relative maximum or minimum) is a point on a graph where the function changes from increasing to decreasing (relative maximum) or decreasing to increasing (relative minimum).

Step2: Analyze each point

• Point \(a\): It's an \(x\)-intercept, the function is increasing before and after? No, before \(a\) the function is going down, after \(a\) it's going up? Wait, no, looking at the graph: from left, the function comes from below, crosses \(a\) (x-intercept), then goes up to \(b\). Wait, no, let's trace the graph: starts from bottom left, goes up to \(a\) (x-intercept), then up to \(b\) (a peak), then down to \(c\) (x-intercept), then down to \(d\) (a valley), then up to \(e\) (x-intercept) and beyond. Wait, no, correction: the graph: left side, comes from below, crosses \(a\) (x-intercept), then rises to \(b\) (so \(b\) is a relative maximum because before \(b\) the function was increasing (from \(a\) to \(b\))? Wait no, wait the direction: when moving from left to right, before \(b\), the function is increasing (going up), after \(b\), it's decreasing (going down). So \(b\) is a relative maximum. Then from \(b\) down to \(d\): after \(b\), function decreases until \(d\), then after \(d\), it increases. So \(d\) is a relative minimum. Now check other points: \(a\) is an x-intercept, the function is increasing through \(a\) (from below, up to \(b\)), so \(a\) is not an extreme. \(c\) is an x-intercept, function is decreasing before (from \(b\) to \(c\)) and increasing after? No, from \(c\) to \(d\) it's decreasing? Wait no, from \(b\) to \(d\), the function goes down (from \(b\) to \(d\)), so at \(c\) (between \(b\) and \(d\)), the function is decreasing through \(c\), so \(c\) is not an extreme. \(e\) is an x-intercept, function is increasing through \(e\) (from \(d\) to \(e\) to beyond), so \(e\) is not an extreme. So only \(b\) (relative max) and \(d\) (relative min) are relative extreme values. So count is 2.

Answer:

2