Question

the given function $f(x)$ is defined on the interval $(-9,9)$ and represented by the graph below. find the x - value(s) of all relative maxima.

Explanation:

Step1: Recall the definition of relative maxima

A relative maximum of a function \(y = f(x)\) is a point \((x_0,f(x_0))\) such that \(f(x_0)\geq f(x)\) for all \(x\) in some open interval containing \(x_0\). Visually, on the graph of the function, it is a "peak" within a local neighborhood.

Step2: Analyze the graph

Looking at the graph of the function \(y = f(x)\) defined on the interval \((-9,9)\). There are two points that satisfy the condition of relative maxima. One is at \(x = 1\) (where the function has a local peak compared to its immediate left - hand and right - hand neighbors in the interval \((-9,9)\)) and the other is at \(x=-7\) (the open - circle at \(x = - 7\) is still a local maximum within the context of the function's behavior in the given interval \((-9,9)\) as we consider the function's values around that non - included point for the local neighborhood concept).

Answer:

\(x=-7\) and \(x = 1\)