To answer this question we need to look closely to the graph of g'(x) and we have to remember that this graph represents the slope of the tangent line of the parent function g(x).
We notice that this function, g'(x), is positive for the intervals:
[tex](-\infty,0)\text{ and (}6,\infty)[/tex]
This means that the graph of the parent function g(x) will increase in those intervals.
We also notice that the graph of the function g'(x) is is negative in the interval:
[tex](0,6)[/tex]
Hence the parent function g(x) will decrease in the interval.
Finally we notice that the graph of g'(x) is zero at x=0 and x=6, this means that the parent function g(x) will have an extreme point at those points; this extreme points could be a maximum, minimum or point of inflection.
Now, to graph the parent function g(x) we first graph the points over the x-axis where we know this function have extreme values:
Now, we know that before the first point the function will increase; bertween the points the function will decrease and after the second point the function will increase. We also know that between the points there's no other extreme point, hence an appropriate sketch of the graph would be:
(Notice how this graph fullfils the condition given by the graph of g'(x) )
Now, the problem gives us another condition, we know that the parent function has to fullfil:
[tex]g(0)=2[/tex]
This means that the graph has to passes thorugh y=2 when x= 0. To do this we translate the graph above two units up. Therefore the graph of the parent function g(x) is:
