Question

question
the function $f(x)$ is graphed below. how many points on the graph represent a relative maximum?
graph of a function with points a, b, c
answer
attempt 1 out of 2

Explanation:

Step1: Define relative maximum

A relative maximum is a point where the function changes from increasing to decreasing, so the point is higher than its immediate neighbors.

Step2: Analyze each point

• Point a: The function is decreasing before a and increasing after? No, before a the function is coming from the top (increasing? Wait, no, the left side of a: the graph is going down to a? Wait, no, the left part of the graph (left of a) is going up? Wait, no, the graph on the left of a: the arrow is going up, so as x approaches a from the left, the function is increasing, and after a, it's decreasing? Wait, no, looking at the graph: point a is on the x-axis. Then from a, the graph goes down to point b (a minimum), then up to point c, then down. Wait, point c: before c, the graph is increasing (from b to c), and after c, it's decreasing. So point c is a relative maximum. Point a: before a, the graph is going up (since the left arrow is up), so as x approaches a from the left, the function is increasing, and after a, it's decreasing? Wait, no, if the graph is coming from the top (left side, arrow up), then at a, the graph crosses the x-axis, then goes down to b. So before a, the function is increasing (since as x increases towards a, the function value increases? Wait, no, if the left side is going up (arrow up), that means as x approaches negative infinity, the function goes to positive infinity, then comes down to a, then down to b. So at point a, the function is decreasing before a? Wait, maybe I messed up. Let's re-express: a relative maximum is a point where f(x) ≥ f(x+h) and f(x) ≥ f(x - h) for small h > 0. So for point c: around c, the points to the left of c are lower (since the graph is going up to c) and points to the right are lower (graph going down from c). So c is a relative maximum. Point a: the points to the left of a: the graph is higher (since it's coming from positive infinity), so f(a) is less than f(a - h) for small h, so a is not a relative maximum. Point b: it's a minimum (since it's lower than neighbors). So only point c is a relative maximum. Wait, but wait, maybe I made a mistake. Wait, the graph: left side (x very negative) goes up (arrow up), then at a, it crosses x-axis, then goes down to b (a minimum), then up to c (on x-axis), then down (arrow down). So from b to c, the graph is increasing (going up to c), then from c, it's decreasing (going down). So c is a relative maximum. How many? 1.

Answer:

1