Question

the above is the graph of the derivative ( f(x) ). how many relative maxima does the function ( f(x) ) have?

Explanation:

Step1: Recall the first - derivative test

A function \(y = f(x)\) has a relative maximum at a point \(x = c\) if \(f^{\prime}(c)=0\) and \(f^{\prime}(x)\) changes sign from positive to negative as \(x\) increases through \(c\).

Step2: Analyze the sign - change of \(f^{\prime}(x)\)

We look for the \(x\) - values where \(f^{\prime}(x)=0\) (the \(x\) - intercepts of \(y = f^{\prime}(x)\)) and then check the sign of \(f^{\prime}(x)\) on either side of these \(x\) - values.
From the graph of \(y = f^{\prime}(x)\), we find the \(x\) - intercepts. Let's assume the critical points (where \(f^{\prime}(x)=0\)) are \(x_1,x_2,\cdots\).
We check the intervals \((-\infty,x_i)\) and \((x_i,\infty)\) for each critical point \(x_i\).
After analyzing the graph (by looking at the behavior of \(y = f^{\prime}(x)\) around its \(x\) - intercepts), we find that there is \(1\) point where \(f^{\prime}(x)\) changes from positive to negative.

Answer:

\(1\)