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)