Question

the function $f(x)$ is graphed below. how many points on the graph represent a relative minimum?

Explanation:

Step1: Recall the definition of a relative minimum

A relative minimum of a function \(y = f(x)\) is a point \((x_0,f(x_0))\) such that \(f(x_0)\leq f(x)\) for all \(x\) in some open interval containing \(x_0\). Graphically, at a relative minimum, the function changes from decreasing to increasing.

Step2: Analyze the given graph

Looking at the graph of \(y = f(x)\):

• At point \(a\), the function crosses the \(x -\)axis. The function is increasing before \(a\) (if we consider the left - hand behavior near \(a\)) and decreasing after \(a\) (if we consider the right - hand behavior near \(a\)) in the local neighborhood.
• At point \(b\), it is a relative maximum since the function changes from increasing to decreasing in its local neighborhood.
• At point \(c\), the function has a horizontal tangent. But the function is decreasing on both sides of \(c\) (in the local neighborhood).

Since there is no point on the graph where the function changes from decreasing to increasing in a local neighborhood, the number of relative minima is \(0\).

Answer:

\(0\)